Trigonometry Examples

$\tan (\frac{7 π}{4} + 0) = \frac{- \cos (x) + \sin (x)}{\cos (x) + \sin (x)}$

Step 1

Rewrite the equation as $\frac{- \cos (x) + \sin (x)}{\cos (x) + \sin (x)} = \tan (\frac{7 π}{4} + 0)$ .

$\frac{- \cos (x) + \sin (x)}{\cos (x) + \sin (x)} = \tan (\frac{7 π}{4} + 0)$

Step 2

Divide each term in the equation by $\cos (x)$ .

$\frac{\frac{- \cos (x) + \sin (x)}{\cos (x) + \sin (x)}}{\cos (x)} = \frac{\tan (\frac{7 π}{4} + 0)}{\cos (x)}$

Step 3

Separate fractions.

$\frac{\frac{- \cos (x) + \sin (x)}{\cos (x) + \sin (x)}}{1} \cdot \frac{1}{\cos (x)} = \frac{\tan (\frac{7 π}{4} + 0)}{\cos (x)}$

Step 4

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\frac{\frac{- \cos (x) + \sin (x)}{\cos (x) + \sin (x)}}{1} \cdot \sec (x) = \frac{\tan (\frac{7 π}{4} + 0)}{\cos (x)}$

Step 5

Divide $\frac{- \cos (x) + \sin (x)}{\cos (x) + \sin (x)}$ by $1$ .

$\frac{- \cos (x) + \sin (x)}{\cos (x) + \sin (x)} \cdot \sec (x) = \frac{\tan (\frac{7 π}{4} + 0)}{\cos (x)}$

Step 6

Combine $\frac{- \cos (x) + \sin (x)}{\cos (x) + \sin (x)}$ and $\sec (x)$ .

$\frac{(- \cos (x) + \sin (x)) \sec (x)}{\cos (x) + \sin (x)} = \frac{\tan (\frac{7 π}{4} + 0)}{\cos (x)}$

Step 7

Factor $- 1$ out of $- \cos (x)$ .

$\frac{(- (\cos (x)) + \sin (x)) \sec (x)}{\cos (x) + \sin (x)} = \frac{\tan (\frac{7 π}{4} + 0)}{\cos (x)}$

Step 8

Factor $- 1$ out of $\sin (x)$ .

$\frac{(- (\cos (x)) - 1 (- \sin (x))) \sec (x)}{\cos (x) + \sin (x)} = \frac{\tan (\frac{7 π}{4} + 0)}{\cos (x)}$

Step 9

Factor $- 1$ out of $- (\cos (x)) - 1 (- \sin (x))$ .

$\frac{- (\cos (x) - \sin (x)) \sec (x)}{\cos (x) + \sin (x)} = \frac{\tan (\frac{7 π}{4} + 0)}{\cos (x)}$

Step 10

Simplify the expression.

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Step 10.1

Rewrite $- (\cos (x) - \sin (x))$ as $- 1 (\cos (x) - \sin (x))$ .

$\frac{- 1 (\cos (x) - \sin (x)) \sec (x)}{\cos (x) + \sin (x)} = \frac{\tan (\frac{7 π}{4} + 0)}{\cos (x)}$

Step 10.2

Move the negative in front of the fraction.

$- \frac{(\cos (x) - \sin (x)) \sec (x)}{\cos (x) + \sin (x)} = \frac{\tan (\frac{7 π}{4} + 0)}{\cos (x)}$

Step 10.3

Reorder factors in $- \frac{(\cos (x) - \sin (x)) \sec (x)}{\cos (x) + \sin (x)}$ .

$- \frac{\sec (x) (\cos (x) - \sin (x))}{\cos (x) + \sin (x)} = \frac{\tan (\frac{7 π}{4} + 0)}{\cos (x)}$

Step 11

Rewrite $\tan (\frac{7 π}{4} + 0)$ in terms of sines and cosines.

$- \frac{\sec (x) (\cos (x) - \sin (x))}{\cos (x) + \sin (x)} = \frac{\frac{\sin (\frac{7 π}{4} + 0)}{\cos (\frac{7 π}{4} + 0)}}{\cos (x)}$

Step 12

Rewrite $\frac{\frac{\sin (\frac{7 π}{4} + 0)}{\cos (\frac{7 π}{4} + 0)}}{\cos (x)}$ as a product.

$- \frac{\sec (x) (\cos (x) - \sin (x))}{\cos (x) + \sin (x)} = \frac{\sin (\frac{7 π}{4} + 0)}{\cos (\frac{7 π}{4} + 0)} \cdot \frac{1}{\cos (x)}$

Step 13

Multiply $\frac{\sin (\frac{7 π}{4} + 0)}{\cos (\frac{7 π}{4} + 0)}$ by $\frac{1}{\cos (x)}$ .

$- \frac{\sec (x) (\cos (x) - \sin (x))}{\cos (x) + \sin (x)} = \frac{\sin (\frac{7 π}{4} + 0)}{\cos (\frac{7 π}{4} + 0) \cos (x)}$

Step 14

Simplify the numerator.

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Step 14.1

Add $\frac{7 π}{4}$ and $0$ .

$- \frac{\sec (x) (\cos (x) - \sin (x))}{\cos (x) + \sin (x)} = \frac{\sin (\frac{7 π}{4})}{\cos (\frac{7 π}{4} + 0) \cos (x)}$

Step 14.2

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the fourth quadrant.

$- \frac{\sec (x) (\cos (x) - \sin (x))}{\cos (x) + \sin (x)} = \frac{- \sin (\frac{π}{4})}{\cos (\frac{7 π}{4} + 0) \cos (x)}$

Step 14.3

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$- \frac{\sec (x) (\cos (x) - \sin (x))}{\cos (x) + \sin (x)} = \frac{- \frac{\sqrt{2}}{2}}{\cos (\frac{7 π}{4} + 0) \cos (x)}$

Step 15

Simplify the denominator.

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Step 15.1

Add $\frac{7 π}{4}$ and $0$ .

$- \frac{\sec (x) (\cos (x) - \sin (x))}{\cos (x) + \sin (x)} = \frac{- \frac{\sqrt{2}}{2}}{\cos (\frac{7 π}{4}) \cos (x)}$

Step 15.2

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$- \frac{\sec (x) (\cos (x) - \sin (x))}{\cos (x) + \sin (x)} = \frac{- \frac{\sqrt{2}}{2}}{\cos (\frac{π}{4}) \cos (x)}$

Step 15.3

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$- \frac{\sec (x) (\cos (x) - \sin (x))}{\cos (x) + \sin (x)} = \frac{- \frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2} \cdot \cos (x)}$

Step 16

Combine $\frac{\sqrt{2}}{2}$ and $\cos (x)$ .

$- \frac{\sec (x) (\cos (x) - \sin (x))}{\cos (x) + \sin (x)} = \frac{- \frac{\sqrt{2}}{2}}{\frac{\sqrt{2} \cos (x)}{2}}$

Step 17

Multiply the numerator by the reciprocal of the denominator.

$- \frac{\sec (x) (\cos (x) - \sin (x))}{\cos (x) + \sin (x)} = - \frac{\sqrt{2}}{2} \cdot \frac{2}{\sqrt{2} \cos (x)}$

Step 18

Cancel the common factor of $\sqrt{2}$ .

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Step 18.1

Move the leading negative in $- \frac{\sqrt{2}}{2}$ into the numerator.

$- \frac{\sec (x) (\cos (x) - \sin (x))}{\cos (x) + \sin (x)} = \frac{- \sqrt{2}}{2} \cdot \frac{2}{\sqrt{2} \cos (x)}$

Step 18.2

Factor $\sqrt{2}$ out of $- \sqrt{2}$ .

$- \frac{\sec (x) (\cos (x) - \sin (x))}{\cos (x) + \sin (x)} = \frac{\sqrt{2} \cdot - 1}{2} \cdot \frac{2}{\sqrt{2} \cos (x)}$

Step 18.3

Factor $\sqrt{2}$ out of $\sqrt{2} \cos (x)$ .

$- \frac{\sec (x) (\cos (x) - \sin (x))}{\cos (x) + \sin (x)} = \frac{\sqrt{2} \cdot - 1}{2} \cdot \frac{2}{\sqrt{2} (\cos (x))}$

Step 18.4

Cancel the common factor.

$- \frac{\sec (x) (\cos (x) - \sin (x))}{\cos (x) + \sin (x)} = \frac{\sqrt{2} \cdot - 1}{2} \cdot \frac{2}{\sqrt{2} \cos (x)}$

Step 18.5

Rewrite the expression.

$- \frac{\sec (x) (\cos (x) - \sin (x))}{\cos (x) + \sin (x)} = \frac{- 1}{2} \cdot \frac{2}{\cos (x)}$

Step 19

Cancel the common factor of $2$ .

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Step 19.1

Cancel the common factor.

$- \frac{\sec (x) (\cos (x) - \sin (x))}{\cos (x) + \sin (x)} = \frac{- 1}{2} \cdot \frac{2}{\cos (x)}$

Step 19.2

Rewrite the expression.

$- \frac{\sec (x) (\cos (x) - \sin (x))}{\cos (x) + \sin (x)} = - \frac{1}{\cos (x)}$

Step 20

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$- \frac{\sec (x) (\cos (x) - \sin (x))}{\cos (x) + \sin (x)} = - \sec (x)$

Step 21

Simplify the left side.

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Step 21.1

Simplify $- \frac{\sec (x) (\cos (x) - \sin (x))}{\cos (x) + \sin (x)}$ .

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Step 21.1.1

Rewrite $\sec (x)$ in terms of sines and cosines.

$- \frac{\frac{1}{\cos (x)} (\cos (x) - \sin (x))}{\cos (x) + \sin (x)} = - \sec (x)$

Step 21.1.2

Factor $\frac{1}{\cos (x)}$ out of $\frac{\frac{1}{\cos (x)} (\cos (x) - \sin (x))}{\cos (x) + \sin (x)}$ .

$- (\frac{1}{\cos (x)} \cdot \frac{\cos (x) - \sin (x)}{\cos (x) + \sin (x)}) = - \sec (x)$

Step 21.1.3

Multiply $\frac{1}{\cos (x)}$ by $\frac{\cos (x) - \sin (x)}{\cos (x) + \sin (x)}$ .

$- \frac{\cos (x) - \sin (x)}{\cos (x) (\cos (x) + \sin (x))} = - \sec (x)$

Step 22

Simplify the right side.

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Step 22.1

Rewrite $\sec (x)$ in terms of sines and cosines.

$- \frac{\cos (x) - \sin (x)}{\cos (x) (\cos (x) + \sin (x))} = - \frac{1}{\cos (x)}$

Step 23

Multiply both sides of the equation by $\cos (x)$ .

$\cos (x) (- \frac{\cos (x) - \sin (x)}{\cos (x) (\cos (x) + \sin (x))}) = \cos (x) (- \frac{1}{\cos (x)})$

Step 24

Rewrite using the commutative property of multiplication.

$- \cos (x) \frac{\cos (x) - \sin (x)}{\cos (x) (\cos (x) + \sin (x))} = \cos (x) (- \frac{1}{\cos (x)})$

Step 25

Cancel the common factor of $\cos (x)$ .

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Step 25.1

Factor $\cos (x)$ out of $- \cos (x)$ .

$\cos (x) \cdot - 1 \frac{\cos (x) - \sin (x)}{\cos (x) (\cos (x) + \sin (x))} = \cos (x) (- \frac{1}{\cos (x)})$

Step 25.2

Cancel the common factor.

$\cos (x) \cdot - 1 \frac{\cos (x) - \sin (x)}{\cos (x) (\cos (x) + \sin (x))} = \cos (x) (- \frac{1}{\cos (x)})$

Step 25.3

Rewrite the expression.

$- \frac{\cos (x) - \sin (x)}{\cos (x) + \sin (x)} = \cos (x) (- \frac{1}{\cos (x)})$

Step 26

Rewrite using the commutative property of multiplication.

$- \frac{\cos (x) - \sin (x)}{\cos (x) + \sin (x)} = - \cos (x) \frac{1}{\cos (x)}$

Step 27

Cancel the common factor of $\cos (x)$ .

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Step 27.1

Factor $\cos (x)$ out of $- \cos (x)$ .

$- \frac{\cos (x) - \sin (x)}{\cos (x) + \sin (x)} = \cos (x) \cdot - 1 \frac{1}{\cos (x)}$

Step 27.2

Cancel the common factor.

$- \frac{\cos (x) - \sin (x)}{\cos (x) + \sin (x)} = \cos (x) \cdot - 1 \frac{1}{\cos (x)}$

Step 27.3

Rewrite the expression.

$- \frac{\cos (x) - \sin (x)}{\cos (x) + \sin (x)} = - 1$

Step 28

Divide each term in the equation by $\cos (x)$ .

$\frac{- \frac{\cos (x) - \sin (x)}{\cos (x) + \sin (x)}}{\cos (x)} = \frac{- 1}{\cos (x)}$

Step 29

Multiply the numerator by the reciprocal of the denominator.

$- \frac{\cos (x) - \sin (x)}{\cos (x) + \sin (x)} \cdot \frac{1}{\cos (x)} = \frac{- 1}{\cos (x)}$

Step 30

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$- \frac{\cos (x) - \sin (x)}{\cos (x) + \sin (x)} \cdot \sec (x) = \frac{- 1}{\cos (x)}$

Step 31

Combine $\sec (x)$ and $\frac{\cos (x) - \sin (x)}{\cos (x) + \sin (x)}$ .

$- \frac{\sec (x) (\cos (x) - \sin (x))}{\cos (x) + \sin (x)} = \frac{- 1}{\cos (x)}$

Step 32

Separate fractions.

$- \frac{\sec (x) (\cos (x) - \sin (x))}{\cos (x) + \sin (x)} = \frac{- 1}{1} \cdot \frac{1}{\cos (x)}$

Step 33

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$- \frac{\sec (x) (\cos (x) - \sin (x))}{\cos (x) + \sin (x)} = \frac{- 1}{1} \cdot \sec (x)$

Step 34

Divide $- 1$ by $1$ .

$- \frac{\sec (x) (\cos (x) - \sin (x))}{\cos (x) + \sin (x)} = - \sec (x)$

Step 35

Simplify the left side.

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Step 35.1

Simplify $- \frac{\sec (x) (\cos (x) - \sin (x))}{\cos (x) + \sin (x)}$ .

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Step 35.1.1

Rewrite $\sec (x)$ in terms of sines and cosines.

$- \frac{\frac{1}{\cos (x)} (\cos (x) - \sin (x))}{\cos (x) + \sin (x)} = - \sec (x)$

Step 35.1.2

Factor $\frac{1}{\cos (x)}$ out of $\frac{\frac{1}{\cos (x)} (\cos (x) - \sin (x))}{\cos (x) + \sin (x)}$ .

$- (\frac{1}{\cos (x)} \cdot \frac{\cos (x) - \sin (x)}{\cos (x) + \sin (x)}) = - \sec (x)$

Step 35.1.3

Multiply $\frac{1}{\cos (x)}$ by $\frac{\cos (x) - \sin (x)}{\cos (x) + \sin (x)}$ .

$- \frac{\cos (x) - \sin (x)}{\cos (x) (\cos (x) + \sin (x))} = - \sec (x)$

Step 36

Simplify the right side.

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Step 36.1

Rewrite $\sec (x)$ in terms of sines and cosines.

$- \frac{\cos (x) - \sin (x)}{\cos (x) (\cos (x) + \sin (x))} = - \frac{1}{\cos (x)}$

Step 37

Multiply both sides of the equation by $\cos (x)$ .

$\cos (x) (- \frac{\cos (x) - \sin (x)}{\cos (x) (\cos (x) + \sin (x))}) = \cos (x) (- \frac{1}{\cos (x)})$

Step 38

Rewrite using the commutative property of multiplication.

$- \cos (x) \frac{\cos (x) - \sin (x)}{\cos (x) (\cos (x) + \sin (x))} = \cos (x) (- \frac{1}{\cos (x)})$

Step 39

Cancel the common factor of $\cos (x)$ .

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Step 39.1

Factor $\cos (x)$ out of $- \cos (x)$ .

$\cos (x) \cdot - 1 \frac{\cos (x) - \sin (x)}{\cos (x) (\cos (x) + \sin (x))} = \cos (x) (- \frac{1}{\cos (x)})$

Step 39.2

Cancel the common factor.

$\cos (x) \cdot - 1 \frac{\cos (x) - \sin (x)}{\cos (x) (\cos (x) + \sin (x))} = \cos (x) (- \frac{1}{\cos (x)})$

Step 39.3

Rewrite the expression.

$- \frac{\cos (x) - \sin (x)}{\cos (x) + \sin (x)} = \cos (x) (- \frac{1}{\cos (x)})$

Step 40

Rewrite using the commutative property of multiplication.

$- \frac{\cos (x) - \sin (x)}{\cos (x) + \sin (x)} = - \cos (x) \frac{1}{\cos (x)}$

Step 41

Cancel the common factor of $\cos (x)$ .

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Step 41.1

Factor $\cos (x)$ out of $- \cos (x)$ .

$- \frac{\cos (x) - \sin (x)}{\cos (x) + \sin (x)} = \cos (x) \cdot - 1 \frac{1}{\cos (x)}$

Step 41.2

Cancel the common factor.

$- \frac{\cos (x) - \sin (x)}{\cos (x) + \sin (x)} = \cos (x) \cdot - 1 \frac{1}{\cos (x)}$

Step 41.3

Rewrite the expression.

$- \frac{\cos (x) - \sin (x)}{\cos (x) + \sin (x)} = - 1$

Step 42

Divide each term in the equation by $\cos (x)$ .

$\frac{- \frac{\cos (x) - \sin (x)}{\cos (x) + \sin (x)}}{\cos (x)} = \frac{- 1}{\cos (x)}$

Step 43

Multiply the numerator by the reciprocal of the denominator.

$- \frac{\cos (x) - \sin (x)}{\cos (x) + \sin (x)} \cdot \frac{1}{\cos (x)} = \frac{- 1}{\cos (x)}$

Step 44

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$- \frac{\cos (x) - \sin (x)}{\cos (x) + \sin (x)} \cdot \sec (x) = \frac{- 1}{\cos (x)}$

Step 45

Combine $\sec (x)$ and $\frac{\cos (x) - \sin (x)}{\cos (x) + \sin (x)}$ .

$- \frac{\sec (x) (\cos (x) - \sin (x))}{\cos (x) + \sin (x)} = \frac{- 1}{\cos (x)}$

Step 46

Separate fractions.

$- \frac{\sec (x) (\cos (x) - \sin (x))}{\cos (x) + \sin (x)} = \frac{- 1}{1} \cdot \frac{1}{\cos (x)}$

Step 47

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$- \frac{\sec (x) (\cos (x) - \sin (x))}{\cos (x) + \sin (x)} = \frac{- 1}{1} \cdot \sec (x)$

Step 48

Divide $- 1$ by $1$ .

$- \frac{\sec (x) (\cos (x) - \sin (x))}{\cos (x) + \sin (x)} = - \sec (x)$

Step 49

Simplify the left side.

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Step 49.1

Simplify $- \frac{\sec (x) (\cos (x) - \sin (x))}{\cos (x) + \sin (x)}$ .

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Step 49.1.1

Rewrite $\sec (x)$ in terms of sines and cosines.

$- \frac{\frac{1}{\cos (x)} (\cos (x) - \sin (x))}{\cos (x) + \sin (x)} = - \sec (x)$

Step 49.1.2

Factor $\frac{1}{\cos (x)}$ out of $\frac{\frac{1}{\cos (x)} (\cos (x) - \sin (x))}{\cos (x) + \sin (x)}$ .

$- (\frac{1}{\cos (x)} \cdot \frac{\cos (x) - \sin (x)}{\cos (x) + \sin (x)}) = - \sec (x)$

Step 49.1.3

Multiply $\frac{1}{\cos (x)}$ by $\frac{\cos (x) - \sin (x)}{\cos (x) + \sin (x)}$ .

$- \frac{\cos (x) - \sin (x)}{\cos (x) (\cos (x) + \sin (x))} = - \sec (x)$

Step 50

Simplify the right side.

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Step 50.1

Rewrite $\sec (x)$ in terms of sines and cosines.

$- \frac{\cos (x) - \sin (x)}{\cos (x) (\cos (x) + \sin (x))} = - \frac{1}{\cos (x)}$

Step 51

Multiply both sides of the equation by $\cos (x)$ .

$\cos (x) (- \frac{\cos (x) - \sin (x)}{\cos (x) (\cos (x) + \sin (x))}) = \cos (x) (- \frac{1}{\cos (x)})$

Step 52

Rewrite using the commutative property of multiplication.

$- \cos (x) \frac{\cos (x) - \sin (x)}{\cos (x) (\cos (x) + \sin (x))} = \cos (x) (- \frac{1}{\cos (x)})$

Step 53

Cancel the common factor of $\cos (x)$ .

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Step 53.1

Factor $\cos (x)$ out of $- \cos (x)$ .

$\cos (x) \cdot - 1 \frac{\cos (x) - \sin (x)}{\cos (x) (\cos (x) + \sin (x))} = \cos (x) (- \frac{1}{\cos (x)})$

Step 53.2

Cancel the common factor.

$\cos (x) \cdot - 1 \frac{\cos (x) - \sin (x)}{\cos (x) (\cos (x) + \sin (x))} = \cos (x) (- \frac{1}{\cos (x)})$

Step 53.3

Rewrite the expression.

$- \frac{\cos (x) - \sin (x)}{\cos (x) + \sin (x)} = \cos (x) (- \frac{1}{\cos (x)})$

Step 54

Rewrite using the commutative property of multiplication.

$- \frac{\cos (x) - \sin (x)}{\cos (x) + \sin (x)} = - \cos (x) \frac{1}{\cos (x)}$

Step 55

Cancel the common factor of $\cos (x)$ .

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Step 55.1

Factor $\cos (x)$ out of $- \cos (x)$ .

$- \frac{\cos (x) - \sin (x)}{\cos (x) + \sin (x)} = \cos (x) \cdot - 1 \frac{1}{\cos (x)}$

Step 55.2

Cancel the common factor.

$- \frac{\cos (x) - \sin (x)}{\cos (x) + \sin (x)} = \cos (x) \cdot - 1 \frac{1}{\cos (x)}$

Step 55.3

Rewrite the expression.

$- \frac{\cos (x) - \sin (x)}{\cos (x) + \sin (x)} = - 1$

Step 56

Divide each term in the equation by $\cos (x)$ .

$\frac{- \frac{\cos (x) - \sin (x)}{\cos (x) + \sin (x)}}{\cos (x)} = \frac{- 1}{\cos (x)}$

Step 57

Multiply the numerator by the reciprocal of the denominator.

$- \frac{\cos (x) - \sin (x)}{\cos (x) + \sin (x)} \cdot \frac{1}{\cos (x)} = \frac{- 1}{\cos (x)}$

Step 58

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$- \frac{\cos (x) - \sin (x)}{\cos (x) + \sin (x)} \cdot \sec (x) = \frac{- 1}{\cos (x)}$

Step 59

Combine $\sec (x)$ and $\frac{\cos (x) - \sin (x)}{\cos (x) + \sin (x)}$ .

$- \frac{\sec (x) (\cos (x) - \sin (x))}{\cos (x) + \sin (x)} = \frac{- 1}{\cos (x)}$

Step 60

Separate fractions.

$- \frac{\sec (x) (\cos (x) - \sin (x))}{\cos (x) + \sin (x)} = \frac{- 1}{1} \cdot \frac{1}{\cos (x)}$

Step 61

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$- \frac{\sec (x) (\cos (x) - \sin (x))}{\cos (x) + \sin (x)} = \frac{- 1}{1} \cdot \sec (x)$

Step 62

Divide $- 1$ by $1$ .

$- \frac{\sec (x) (\cos (x) - \sin (x))}{\cos (x) + \sin (x)} = - \sec (x)$

Step 63

Simplify the left side.

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Step 63.1

Simplify $- \frac{\sec (x) (\cos (x) - \sin (x))}{\cos (x) + \sin (x)}$ .

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Step 63.1.1

Rewrite $\sec (x)$ in terms of sines and cosines.

$- \frac{\frac{1}{\cos (x)} (\cos (x) - \sin (x))}{\cos (x) + \sin (x)} = - \sec (x)$

Step 63.1.2

Factor $\frac{1}{\cos (x)}$ out of $\frac{\frac{1}{\cos (x)} (\cos (x) - \sin (x))}{\cos (x) + \sin (x)}$ .

$- (\frac{1}{\cos (x)} \cdot \frac{\cos (x) - \sin (x)}{\cos (x) + \sin (x)}) = - \sec (x)$

Step 63.1.3

Multiply $\frac{1}{\cos (x)}$ by $\frac{\cos (x) - \sin (x)}{\cos (x) + \sin (x)}$ .

$- \frac{\cos (x) - \sin (x)}{\cos (x) (\cos (x) + \sin (x))} = - \sec (x)$

Step 64

Simplify the right side.

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Step 64.1

Rewrite $\sec (x)$ in terms of sines and cosines.

$- \frac{\cos (x) - \sin (x)}{\cos (x) (\cos (x) + \sin (x))} = - \frac{1}{\cos (x)}$

Step 65

Multiply both sides of the equation by $\cos (x)$ .

$\cos (x) (- \frac{\cos (x) - \sin (x)}{\cos (x) (\cos (x) + \sin (x))}) = \cos (x) (- \frac{1}{\cos (x)})$

Step 66

Rewrite using the commutative property of multiplication.

$- \cos (x) \frac{\cos (x) - \sin (x)}{\cos (x) (\cos (x) + \sin (x))} = \cos (x) (- \frac{1}{\cos (x)})$

Step 67

Cancel the common factor of $\cos (x)$ .

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Step 67.1

Factor $\cos (x)$ out of $- \cos (x)$ .

$\cos (x) \cdot - 1 \frac{\cos (x) - \sin (x)}{\cos (x) (\cos (x) + \sin (x))} = \cos (x) (- \frac{1}{\cos (x)})$

Step 67.2

Cancel the common factor.

$\cos (x) \cdot - 1 \frac{\cos (x) - \sin (x)}{\cos (x) (\cos (x) + \sin (x))} = \cos (x) (- \frac{1}{\cos (x)})$

Step 67.3

Rewrite the expression.

$- \frac{\cos (x) - \sin (x)}{\cos (x) + \sin (x)} = \cos (x) (- \frac{1}{\cos (x)})$

Step 68

Rewrite using the commutative property of multiplication.

$- \frac{\cos (x) - \sin (x)}{\cos (x) + \sin (x)} = - \cos (x) \frac{1}{\cos (x)}$

Step 69

Cancel the common factor of $\cos (x)$ .

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Step 69.1

Factor $\cos (x)$ out of $- \cos (x)$ .

$- \frac{\cos (x) - \sin (x)}{\cos (x) + \sin (x)} = \cos (x) \cdot - 1 \frac{1}{\cos (x)}$

Step 69.2

Cancel the common factor.

$- \frac{\cos (x) - \sin (x)}{\cos (x) + \sin (x)} = \cos (x) \cdot - 1 \frac{1}{\cos (x)}$

Step 69.3

Rewrite the expression.

$- \frac{\cos (x) - \sin (x)}{\cos (x) + \sin (x)} = - 1$

Step 70

Divide each term in the equation by $\cos (x)$ .

$\frac{- \frac{\cos (x) - \sin (x)}{\cos (x) + \sin (x)}}{\cos (x)} = \frac{- 1}{\cos (x)}$

Step 71

Multiply the numerator by the reciprocal of the denominator.

$- \frac{\cos (x) - \sin (x)}{\cos (x) + \sin (x)} \cdot \frac{1}{\cos (x)} = \frac{- 1}{\cos (x)}$

Step 72

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$- \frac{\cos (x) - \sin (x)}{\cos (x) + \sin (x)} \cdot \sec (x) = \frac{- 1}{\cos (x)}$

Step 73

Combine $\sec (x)$ and $\frac{\cos (x) - \sin (x)}{\cos (x) + \sin (x)}$ .

$- \frac{\sec (x) (\cos (x) - \sin (x))}{\cos (x) + \sin (x)} = \frac{- 1}{\cos (x)}$

Step 74

Separate fractions.

$- \frac{\sec (x) (\cos (x) - \sin (x))}{\cos (x) + \sin (x)} = \frac{- 1}{1} \cdot \frac{1}{\cos (x)}$

Step 75

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$- \frac{\sec (x) (\cos (x) - \sin (x))}{\cos (x) + \sin (x)} = \frac{- 1}{1} \cdot \sec (x)$

Step 76

Divide $- 1$ by $1$ .

$- \frac{\sec (x) (\cos (x) - \sin (x))}{\cos (x) + \sin (x)} = - \sec (x)$

Step 77

Simplify the left side.

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Step 77.1

Simplify $- \frac{\sec (x) (\cos (x) - \sin (x))}{\cos (x) + \sin (x)}$ .

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Step 77.1.1

Rewrite $\sec (x)$ in terms of sines and cosines.

$- \frac{\frac{1}{\cos (x)} (\cos (x) - \sin (x))}{\cos (x) + \sin (x)} = - \sec (x)$

Step 77.1.2

Factor $\frac{1}{\cos (x)}$ out of $\frac{\frac{1}{\cos (x)} (\cos (x) - \sin (x))}{\cos (x) + \sin (x)}$ .

$- (\frac{1}{\cos (x)} \cdot \frac{\cos (x) - \sin (x)}{\cos (x) + \sin (x)}) = - \sec (x)$

Step 77.1.3

Multiply $\frac{1}{\cos (x)}$ by $\frac{\cos (x) - \sin (x)}{\cos (x) + \sin (x)}$ .

$- \frac{\cos (x) - \sin (x)}{\cos (x) (\cos (x) + \sin (x))} = - \sec (x)$

Step 78

Divide each term in the equation by $\cos (x)$ .

$\frac{- \frac{\cos (x) - \sin (x)}{\cos (x) (\cos (x) + \sin (x))}}{\cos (x)} = \frac{- \sec (x)}{\cos (x)}$

Step 79

Multiply the numerator by the reciprocal of the denominator.

$- \frac{\cos (x) - \sin (x)}{\cos (x) (\cos (x) + \sin (x))} \cdot \frac{1}{\cos (x)} = \frac{- \sec (x)}{\cos (x)}$

Step 80

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$- \frac{\cos (x) - \sin (x)}{\cos (x) (\cos (x) + \sin (x))} \cdot \sec (x) = \frac{- \sec (x)}{\cos (x)}$

Step 81

Combine $\sec (x)$ and $\frac{\cos (x) - \sin (x)}{\cos (x) (\cos (x) + \sin (x))}$ .

$- \frac{\sec (x) (\cos (x) - \sin (x))}{\cos (x) (\cos (x) + \sin (x))} = \frac{- \sec (x)}{\cos (x)}$

Step 82

Separate fractions.

$- (\frac{\cos (x) - \sin (x)}{\cos (x) + \sin (x)} \cdot \frac{\sec (x)}{\cos (x)}) = \frac{- \sec (x)}{\cos (x)}$

Step 83

Rewrite $\sec (x)$ in terms of sines and cosines.

$- (\frac{\cos (x) - \sin (x)}{\cos (x) + \sin (x)} \cdot \frac{\frac{1}{\cos (x)}}{\cos (x)}) = \frac{- \sec (x)}{\cos (x)}$

Step 84

Rewrite $\frac{\frac{1}{\cos (x)}}{\cos (x)}$ as a product.

$- (\frac{\cos (x) - \sin (x)}{\cos (x) + \sin (x)} \cdot (\frac{1}{\cos (x)} \cdot \frac{1}{\cos (x)})) = \frac{- \sec (x)}{\cos (x)}$

Step 85

Simplify.

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Step 85.1

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$- (\frac{\cos (x) - \sin (x)}{\cos (x) + \sin (x)} \cdot (\sec (x) (\frac{1}{\cos (x)}))) = \frac{- \sec (x)}{\cos (x)}$

Step 85.2

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$- (\frac{\cos (x) - \sin (x)}{\cos (x) + \sin (x)} \cdot (\sec (x) \sec (x))) = \frac{- \sec (x)}{\cos (x)}$

Step 85.3

Raise $\sec (x)$ to the power of $1$ .

$- (\frac{\cos (x) - \sin (x)}{\cos (x) + \sin (x)} \cdot (\sec (x) \sec (x))) = \frac{- \sec (x)}{\cos (x)}$

Step 85.4

Raise $\sec (x)$ to the power of $1$ .

$- (\frac{\cos (x) - \sin (x)}{\cos (x) + \sin (x)} \cdot (\sec (x) \sec (x))) = \frac{- \sec (x)}{\cos (x)}$

Step 85.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- (\frac{\cos (x) - \sin (x)}{\cos (x) + \sin (x)} \cdot {\sec (x)}^{1 + 1}) = \frac{- \sec (x)}{\cos (x)}$

Step 85.6

Add $1$ and $1$ .

$- (\frac{\cos (x) - \sin (x)}{\cos (x) + \sin (x)} \cdot \sec^{2} (x)) = \frac{- \sec (x)}{\cos (x)}$

Step 86

Combine $\frac{\cos (x) - \sin (x)}{\cos (x) + \sin (x)}$ and $\sec^{2} (x)$ .

$- \frac{(\cos (x) - \sin (x)) \sec^{2} (x)}{\cos (x) + \sin (x)} = \frac{- \sec (x)}{\cos (x)}$

Step 87

Reorder factors in $- \frac{(\cos (x) - \sin (x)) \sec^{2} (x)}{\cos (x) + \sin (x)}$ .

$- \frac{\sec^{2} (x) (\cos (x) - \sin (x))}{\cos (x) + \sin (x)} = \frac{- \sec (x)}{\cos (x)}$

Step 88

Separate fractions.

$- \frac{\sec^{2} (x) (\cos (x) - \sin (x))}{\cos (x) + \sin (x)} = \frac{- 1}{1} \cdot \frac{\sec (x)}{\cos (x)}$

Step 89

Rewrite $\sec (x)$ in terms of sines and cosines.

$- \frac{\sec^{2} (x) (\cos (x) - \sin (x))}{\cos (x) + \sin (x)} = \frac{- 1}{1} \cdot \frac{\frac{1}{\cos (x)}}{\cos (x)}$

Step 90

Rewrite $\frac{\frac{1}{\cos (x)}}{\cos (x)}$ as a product.

$- \frac{\sec^{2} (x) (\cos (x) - \sin (x))}{\cos (x) + \sin (x)} = \frac{- 1}{1} \cdot (\frac{1}{\cos (x)} \cdot \frac{1}{\cos (x)})$

Step 91

Combine fractions.

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Step 91.1

Multiply $\frac{1}{\cos (x)}$ by $\frac{1}{\cos (x)}$ .

$- \frac{\sec^{2} (x) (\cos (x) - \sin (x))}{\cos (x) + \sin (x)} = \frac{- 1}{1} \cdot \frac{1}{\cos (x) \cos (x)}$

Step 91.2

Divide $- 1$ by $1$ .

$- \frac{\sec^{2} (x) (\cos (x) - \sin (x))}{\cos (x) + \sin (x)} = - \frac{1}{\cos (x) \cos (x)}$

Step 92

Simplify the denominator.

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Step 92.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- \frac{\sec^{2} (x) (\cos (x) - \sin (x))}{\cos (x) + \sin (x)} = - \frac{1}{{\cos (x)}^{1 + 1}}$

Step 92.2

Add $1$ and $1$ .

$- \frac{\sec^{2} (x) (\cos (x) - \sin (x))}{\cos (x) + \sin (x)} = - \frac{1}{\cos^{2} (x)}$

Step 93

Simplify with factoring out.

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Step 93.1

Rewrite $1$ as $1^{2}$ .

$- \frac{\sec^{2} (x) (\cos (x) - \sin (x))}{\cos (x) + \sin (x)} = - \frac{1^{2}}{\cos^{2} (x)}$

Step 93.2

Rewrite $\frac{1^{2}}{\cos^{2} (x)}$ as ${(\frac{1}{\cos (x)})}^{2}$ .

$- \frac{\sec^{2} (x) (\cos (x) - \sin (x))}{\cos (x) + \sin (x)} = - {(\frac{1}{\cos (x)})}^{2}$

Step 94

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$- \frac{\sec^{2} (x) (\cos (x) - \sin (x))}{\cos (x) + \sin (x)} = - \sec^{2} (x)$

Step 95

Replace the $\sec^{2} (x)$ with $1 + \tan^{2} (x)$ based on the $\tan^{2} (x) + 1 = \sec^{2} (x)$ identity.

$\frac{(1 + \tan^{2} (x)) (\cos (x) - \sin (x))}{\cos (x) + \sin (x)} = - \sec^{2} (x)$

Step 96

Multiply both sides by $\cos (x) + \sin (x)$ .

$\frac{(1 + \tan^{2} (x)) (\cos (x) - \sin (x))}{\cos (x) + \sin (x)} (\cos (x) + \sin (x)) = - \sec^{2} (x) (\cos (x) + \sin (x))$

Step 97

Simplify.

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Step 97.1

Simplify the left side.

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Step 97.1.1

Simplify $\frac{(1 + \tan^{2} (x)) (\cos (x) - \sin (x))}{\cos (x) + \sin (x)} (\cos (x) + \sin (x))$ .

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Step 97.1.1.1

Cancel the common factor of $\cos (x) + \sin (x)$ .

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Step 97.1.1.1.1

Cancel the common factor.

$\frac{(1 + \tan^{2} (x)) (\cos (x) - \sin (x))}{\cos (x) + \sin (x)} (\cos (x) + \sin (x)) = - \sec^{2} (x) (\cos (x) + \sin (x))$

Step 97.1.1.1.2

Rewrite the expression.

$(1 + \tan^{2} (x)) (\cos (x) - \sin (x)) = - \sec^{2} (x) (\cos (x) + \sin (x))$

Step 97.1.1.2

Rearrange terms.

$(\tan^{2} (x) + 1) (\cos (x) - \sin (x)) = - \sec^{2} (x) (\cos (x) + \sin (x))$

Step 97.1.1.3

Apply pythagorean identity.

$\sec^{2} (x) (\cos (x) - \sin (x)) = - \sec^{2} (x) (\cos (x) + \sin (x))$

Step 97.1.1.4

Rewrite $\sec (x)$ in terms of sines and cosines.

${(\frac{1}{\cos (x)})}^{2} (\cos (x) - \sin (x)) = - \sec^{2} (x) (\cos (x) + \sin (x))$

Step 97.1.1.5

Simplify terms.

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Step 97.1.1.5.1

Apply the product rule to $\frac{1}{\cos (x)}$ .

$\frac{1^{2}}{\cos^{2} (x)} (\cos (x) - \sin (x)) = - \sec^{2} (x) (\cos (x) + \sin (x))$

Step 97.1.1.5.2

One to any power is one.

$\frac{1}{\cos^{2} (x)} (\cos (x) - \sin (x)) = - \sec^{2} (x) (\cos (x) + \sin (x))$

Step 97.1.1.5.3

Apply the distributive property.

$\frac{1}{\cos^{2} (x)} \cos (x) + \frac{1}{\cos^{2} (x)} (- \sin (x)) = - \sec^{2} (x) (\cos (x) + \sin (x))$

Step 97.1.1.5.4

Cancel the common factor of $\cos (x)$ .

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Step 97.1.1.5.4.1

Factor $\cos (x)$ out of $\cos^{2} (x)$ .

$\frac{1}{\cos (x) \cos (x)} \cos (x) + \frac{1}{\cos^{2} (x)} (- \sin (x)) = - \sec^{2} (x) (\cos (x) + \sin (x))$

Step 97.1.1.5.4.2

Cancel the common factor.

$\frac{1}{\cos (x) \cos (x)} \cos (x) + \frac{1}{\cos^{2} (x)} (- \sin (x)) = - \sec^{2} (x) (\cos (x) + \sin (x))$

Step 97.1.1.5.4.3

Rewrite the expression.

$\frac{1}{\cos (x)} + \frac{1}{\cos^{2} (x)} (- \sin (x)) = - \sec^{2} (x) (\cos (x) + \sin (x))$

Step 97.1.1.5.5

Rewrite using the commutative property of multiplication.

$\frac{1}{\cos (x)} - \frac{1}{\cos^{2} (x)} \sin (x) = - \sec^{2} (x) (\cos (x) + \sin (x))$

Step 97.1.1.5.6

Combine $\sin (x)$ and $\frac{1}{\cos^{2} (x)}$ .

$\frac{1}{\cos (x)} - \frac{\sin (x)}{\cos^{2} (x)} = - \sec^{2} (x) (\cos (x) + \sin (x))$

Step 97.1.1.6

Simplify each term.

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Step 97.1.1.6.1

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\sec (x) - \frac{\sin (x)}{\cos^{2} (x)} = - \sec^{2} (x) (\cos (x) + \sin (x))$

Step 97.1.1.6.2

Factor $\cos (x)$ out of $\cos^{2} (x)$ .

$\sec (x) - \frac{\sin (x)}{\cos (x) \cos (x)} = - \sec^{2} (x) (\cos (x) + \sin (x))$

Step 97.1.1.6.3

Separate fractions.

$\sec (x) - (\frac{1}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)}) = - \sec^{2} (x) (\cos (x) + \sin (x))$

Step 97.1.1.6.4

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\sec (x) - (\frac{1}{\cos (x)} \tan (x)) = - \sec^{2} (x) (\cos (x) + \sin (x))$

Step 97.1.1.6.5

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\sec (x) - \sec (x) \tan (x) = - \sec^{2} (x) (\cos (x) + \sin (x))$

Step 97.2

Simplify the right side.

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Step 97.2.1

Simplify $- \sec^{2} (x) (\cos (x) + \sin (x))$ .

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Step 97.2.1.1

Rewrite $\sec (x)$ in terms of sines and cosines.

$\sec (x) - \sec (x) \tan (x) = - {(\frac{1}{\cos (x)})}^{2} (\cos (x) + \sin (x))$

Step 97.2.1.2

Simplify terms.

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Step 97.2.1.2.1

Apply the product rule to $\frac{1}{\cos (x)}$ .

$\sec (x) - \sec (x) \tan (x) = - \frac{1^{2}}{\cos^{2} (x)} (\cos (x) + \sin (x))$

Step 97.2.1.2.2

One to any power is one.

$\sec (x) - \sec (x) \tan (x) = - \frac{1}{\cos^{2} (x)} (\cos (x) + \sin (x))$

Step 97.2.1.2.3

Apply the distributive property.

$\sec (x) - \sec (x) \tan (x) = - \frac{1}{\cos^{2} (x)} \cos (x) - \frac{1}{\cos^{2} (x)} \sin (x)$

Step 97.2.1.2.4

Cancel the common factor of $\cos (x)$ .

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Step 97.2.1.2.4.1

Move the leading negative in $- \frac{1}{\cos^{2} (x)}$ into the numerator.

$\sec (x) - \sec (x) \tan (x) = \frac{- 1}{\cos^{2} (x)} \cos (x) - \frac{1}{\cos^{2} (x)} \sin (x)$

Step 97.2.1.2.4.2

Factor $\cos (x)$ out of $\cos^{2} (x)$ .

$\sec (x) - \sec (x) \tan (x) = \frac{- 1}{\cos (x) \cos (x)} \cos (x) - \frac{1}{\cos^{2} (x)} \sin (x)$

Step 97.2.1.2.4.3

Cancel the common factor.

$\sec (x) - \sec (x) \tan (x) = \frac{- 1}{\cos (x) \cos (x)} \cos (x) - \frac{1}{\cos^{2} (x)} \sin (x)$

Step 97.2.1.2.4.4

Rewrite the expression.

$\sec (x) - \sec (x) \tan (x) = \frac{- 1}{\cos (x)} - \frac{1}{\cos^{2} (x)} \sin (x)$

Step 97.2.1.2.5

Combine $\sin (x)$ and $\frac{1}{\cos^{2} (x)}$ .

$\sec (x) - \sec (x) \tan (x) = \frac{- 1}{\cos (x)} - \frac{\sin (x)}{\cos^{2} (x)}$

Step 97.2.1.2.6

Move the negative in front of the fraction.

$\sec (x) - \sec (x) \tan (x) = - \frac{1}{\cos (x)} - \frac{\sin (x)}{\cos^{2} (x)}$

Step 97.2.1.3

Simplify each term.

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Step 97.2.1.3.1

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\sec (x) - \sec (x) \tan (x) = - \sec (x) - \frac{\sin (x)}{\cos^{2} (x)}$

Step 97.2.1.3.2

Factor $\cos (x)$ out of $\cos^{2} (x)$ .

$\sec (x) - \sec (x) \tan (x) = - \sec (x) - \frac{\sin (x)}{\cos (x) \cos (x)}$

Step 97.2.1.3.3

Separate fractions.

$\sec (x) - \sec (x) \tan (x) = - \sec (x) - (\frac{1}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)})$

Step 97.2.1.3.4

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\sec (x) - \sec (x) \tan (x) = - \sec (x) - (\frac{1}{\cos (x)} \tan (x))$

Step 97.2.1.3.5

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\sec (x) - \sec (x) \tan (x) = - \sec (x) - \sec (x) \tan (x)$

Step 98

Solve for $x$ .

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Step 98.1

Simplify the left side.

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Step 98.1.1

Simplify each term.

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Step 98.1.1.1

Rewrite $\sec (x)$ in terms of sines and cosines.

$\frac{1}{\cos (x)} - \sec (x) \tan (x) = - \sec (x) - \sec (x) \tan (x)$

Step 98.1.1.2

Rewrite $\sec (x)$ in terms of sines and cosines.

$\frac{1}{\cos (x)} - \frac{1}{\cos (x)} \tan (x) = - \sec (x) - \sec (x) \tan (x)$

Step 98.1.1.3

Rewrite $\tan (x)$ in terms of sines and cosines.

$\frac{1}{\cos (x)} - \frac{1}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)} = - \sec (x) - \sec (x) \tan (x)$

Step 98.1.1.4

Multiply $- \frac{1}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)}$ .

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Step 98.1.1.4.1

Multiply $\frac{\sin (x)}{\cos (x)}$ by $\frac{1}{\cos (x)}$ .

$\frac{1}{\cos (x)} - \frac{\sin (x)}{\cos (x) \cos (x)} = - \sec (x) - \sec (x) \tan (x)$

Step 98.1.1.4.2

Raise $\cos (x)$ to the power of $1$ .

$\frac{1}{\cos (x)} - \frac{\sin (x)}{\cos^{1} (x) \cos (x)} = - \sec (x) - \sec (x) \tan (x)$

Step 98.1.1.4.3

Raise $\cos (x)$ to the power of $1$ .

$\frac{1}{\cos (x)} - \frac{\sin (x)}{\cos^{1} (x) \cos^{1} (x)} = - \sec (x) - \sec (x) \tan (x)$

Step 98.1.1.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1}{\cos (x)} - \frac{\sin (x)}{{\cos (x)}^{1 + 1}} = - \sec (x) - \sec (x) \tan (x)$

Step 98.1.1.4.5

Add $1$ and $1$ .

$\frac{1}{\cos (x)} - \frac{\sin (x)}{\cos^{2} (x)} = - \sec (x) - \sec (x) \tan (x)$

Step 98.2

Simplify the right side.

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Step 98.2.1

Simplify each term.

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Step 98.2.1.1

Rewrite $\sec (x)$ in terms of sines and cosines.

$\frac{1}{\cos (x)} - \frac{\sin (x)}{\cos^{2} (x)} = - \frac{1}{\cos (x)} - \sec (x) \tan (x)$

Step 98.2.1.2

Rewrite $\sec (x)$ in terms of sines and cosines.

$\frac{1}{\cos (x)} - \frac{\sin (x)}{\cos^{2} (x)} = - \frac{1}{\cos (x)} - \frac{1}{\cos (x)} \tan (x)$

Step 98.2.1.3

Rewrite $\tan (x)$ in terms of sines and cosines.

$\frac{1}{\cos (x)} - \frac{\sin (x)}{\cos^{2} (x)} = - \frac{1}{\cos (x)} - \frac{1}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)}$

Step 98.2.1.4

Multiply $- \frac{1}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)}$ .

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Step 98.2.1.4.1

Multiply $\frac{\sin (x)}{\cos (x)}$ by $\frac{1}{\cos (x)}$ .

$\frac{1}{\cos (x)} - \frac{\sin (x)}{\cos^{2} (x)} = - \frac{1}{\cos (x)} - \frac{\sin (x)}{\cos (x) \cos (x)}$

Step 98.2.1.4.2

Raise $\cos (x)$ to the power of $1$ .

$\frac{1}{\cos (x)} - \frac{\sin (x)}{\cos^{2} (x)} = - \frac{1}{\cos (x)} - \frac{\sin (x)}{\cos^{1} (x) \cos (x)}$

Step 98.2.1.4.3

Raise $\cos (x)$ to the power of $1$ .

$\frac{1}{\cos (x)} - \frac{\sin (x)}{\cos^{2} (x)} = - \frac{1}{\cos (x)} - \frac{\sin (x)}{\cos^{1} (x) \cos^{1} (x)}$

Step 98.2.1.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1}{\cos (x)} - \frac{\sin (x)}{\cos^{2} (x)} = - \frac{1}{\cos (x)} - \frac{\sin (x)}{{\cos (x)}^{1 + 1}}$

Step 98.2.1.4.5

Add $1$ and $1$ .

$\frac{1}{\cos (x)} - \frac{\sin (x)}{\cos^{2} (x)} = - \frac{1}{\cos (x)} - \frac{\sin (x)}{\cos^{2} (x)}$

Step 98.3

Multiply both sides of the equation by $\cos (x)$ .

$\cos (x) (\frac{1}{\cos (x)} - \frac{\sin (x)}{\cos^{2} (x)}) = \cos (x) (- \frac{1}{\cos (x)} - \frac{\sin (x)}{\cos^{2} (x)})$

Step 98.4

Apply the distributive property.

$\cos (x) \frac{1}{\cos (x)} + \cos (x) (- \frac{\sin (x)}{\cos^{2} (x)}) = \cos (x) (- \frac{1}{\cos (x)} - \frac{\sin (x)}{\cos^{2} (x)})$

Step 98.5

Cancel the common factor of $\cos (x)$ .

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Step 98.5.1

Cancel the common factor.

$\cos (x) \frac{1}{\cos (x)} + \cos (x) (- \frac{\sin (x)}{\cos^{2} (x)}) = \cos (x) (- \frac{1}{\cos (x)} - \frac{\sin (x)}{\cos^{2} (x)})$

Step 98.5.2

Rewrite the expression.

$1 + \cos (x) (- \frac{\sin (x)}{\cos^{2} (x)}) = \cos (x) (- \frac{1}{\cos (x)} - \frac{\sin (x)}{\cos^{2} (x)})$

Step 98.6

Rewrite using the commutative property of multiplication.

$1 - \cos (x) \frac{\sin (x)}{\cos^{2} (x)} = \cos (x) (- \frac{1}{\cos (x)} - \frac{\sin (x)}{\cos^{2} (x)})$

Step 98.7

Cancel the common factor of $\cos (x)$ .

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Step 98.7.1

Factor $\cos (x)$ out of $- \cos (x)$ .

$1 + \cos (x) \cdot - 1 \frac{\sin (x)}{\cos^{2} (x)} = \cos (x) (- \frac{1}{\cos (x)} - \frac{\sin (x)}{\cos^{2} (x)})$

Step 98.7.2

Factor $\cos (x)$ out of $\cos^{2} (x)$ .

$1 + \cos (x) \cdot - 1 \frac{\sin (x)}{\cos (x) \cos (x)} = \cos (x) (- \frac{1}{\cos (x)} - \frac{\sin (x)}{\cos^{2} (x)})$

Step 98.7.3

Cancel the common factor.

$1 + \cos (x) \cdot - 1 \frac{\sin (x)}{\cos (x) \cos (x)} = \cos (x) (- \frac{1}{\cos (x)} - \frac{\sin (x)}{\cos^{2} (x)})$

Step 98.7.4

Rewrite the expression.

$1 - \frac{\sin (x)}{\cos (x)} = \cos (x) (- \frac{1}{\cos (x)} - \frac{\sin (x)}{\cos^{2} (x)})$

Step 98.8

Apply the distributive property.

$1 - \frac{\sin (x)}{\cos (x)} = \cos (x) (- \frac{1}{\cos (x)}) + \cos (x) (- \frac{\sin (x)}{\cos^{2} (x)})$

Step 98.9

Rewrite using the commutative property of multiplication.

$1 - \frac{\sin (x)}{\cos (x)} = - \cos (x) \frac{1}{\cos (x)} + \cos (x) (- \frac{\sin (x)}{\cos^{2} (x)})$

Step 98.10

Rewrite using the commutative property of multiplication.

$1 - \frac{\sin (x)}{\cos (x)} = - \cos (x) \frac{1}{\cos (x)} - \cos (x) \frac{\sin (x)}{\cos^{2} (x)}$

Step 98.11

Simplify each term.

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Step 98.11.1

Cancel the common factor of $\cos (x)$ .

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Step 98.11.1.1

Factor $\cos (x)$ out of $- \cos (x)$ .

$1 - \frac{\sin (x)}{\cos (x)} = \cos (x) \cdot - 1 \frac{1}{\cos (x)} - \cos (x) \frac{\sin (x)}{\cos^{2} (x)}$

Step 98.11.1.2

Cancel the common factor.

$1 - \frac{\sin (x)}{\cos (x)} = \cos (x) \cdot - 1 \frac{1}{\cos (x)} - \cos (x) \frac{\sin (x)}{\cos^{2} (x)}$

Step 98.11.1.3

Rewrite the expression.

$1 - \frac{\sin (x)}{\cos (x)} = - 1 - \cos (x) \frac{\sin (x)}{\cos^{2} (x)}$

Step 98.11.2

Cancel the common factor of $\cos (x)$ .

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Step 98.11.2.1

Factor $\cos (x)$ out of $- \cos (x)$ .

$1 - \frac{\sin (x)}{\cos (x)} = - 1 + \cos (x) \cdot - 1 \frac{\sin (x)}{\cos^{2} (x)}$

Step 98.11.2.2

Factor $\cos (x)$ out of $\cos^{2} (x)$ .

$1 - \frac{\sin (x)}{\cos (x)} = - 1 + \cos (x) \cdot - 1 \frac{\sin (x)}{\cos (x) \cos (x)}$

Step 98.11.2.3

Cancel the common factor.

$1 - \frac{\sin (x)}{\cos (x)} = - 1 + \cos (x) \cdot - 1 \frac{\sin (x)}{\cos (x) \cos (x)}$

Step 98.11.2.4

Rewrite the expression.

$1 - \frac{\sin (x)}{\cos (x)} = - 1 - \frac{\sin (x)}{\cos (x)}$

Step 98.12

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$1 - \tan (x) = - 1 - \frac{\sin (x)}{\cos (x)}$

Step 98.13

Move all terms containing $\tan (x)$ to the left side of the equation.

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Step 98.13.1

Add $\tan (x)$ to both sides of the equation.

$- \tan (x) + \tan (x) = - 2$

Step 98.13.2

Add $- \tan (x)$ and $\tan (x)$ .

$0 = - 2$

Step 98.14

Since $0 \neq - 2$ , there are no solutions.

No solution