Trigonometry Examples

$\sin (x) - 8 = \cos (x) - 8$

Step 1

Subtract $\cos (x)$ from both sides of the equation.

$\sin (x) - 8 - \cos (x) = - 8$

Step 2

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

$\frac{\sin (x)}{\cos (x)} + \frac{- 8}{\cos (x)} + \frac{- \cos (x)}{\cos (x)} = \frac{- 8}{\cos (x)}$

Step 3

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

$\tan (x) + \frac{- 8}{\cos (x)} + \frac{- \cos (x)}{\cos (x)} = \frac{- 8}{\cos (x)}$

Step 4

Separate fractions.

$\tan (x) + \frac{- 8}{1} \cdot \frac{1}{\cos (x)} + \frac{- \cos (x)}{\cos (x)} = \frac{- 8}{\cos (x)}$

Step 5

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

$\tan (x) + \frac{- 8}{1} \cdot \sec (x) + \frac{- \cos (x)}{\cos (x)} = \frac{- 8}{\cos (x)}$

Step 6

Divide $- 8$ by $1$ .

$\tan (x) - 8 \sec (x) + \frac{- \cos (x)}{\cos (x)} = \frac{- 8}{\cos (x)}$

Step 7

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

Tap for more steps...

Step 7.1

Cancel the common factor.

$\tan (x) - 8 \sec (x) + \frac{- \cos (x)}{\cos (x)} = \frac{- 8}{\cos (x)}$

Step 7.2

Divide $- 1$ by $1$ .

$\tan (x) - 8 \sec (x) - 1 = \frac{- 8}{\cos (x)}$

Step 8

Separate fractions.

$\tan (x) - 8 \sec (x) - 1 = \frac{- 8}{1} \cdot \frac{1}{\cos (x)}$

Step 9

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

$\tan (x) - 8 \sec (x) - 1 = \frac{- 8}{1} \cdot \sec (x)$

Step 10

Divide $- 8$ by $1$ .

$\tan (x) - 8 \sec (x) - 1 = - 8 \sec (x)$

Step 11

Simplify the left side.

Tap for more steps...

Step 11.1

Simplify each term.

Tap for more steps...

Step 11.1.1

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

$\frac{\sin (x)}{\cos (x)} - 8 \sec (x) - 1 = - 8 \sec (x)$

Step 11.1.2

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

$\frac{\sin (x)}{\cos (x)} - 8 \frac{1}{\cos (x)} - 1 = - 8 \sec (x)$

Step 11.1.3

Combine $- 8$ and $\frac{1}{\cos (x)}$ .

$\frac{\sin (x)}{\cos (x)} + \frac{- 8}{\cos (x)} - 1 = - 8 \sec (x)$

Step 11.1.4

Move the negative in front of the fraction.

$\frac{\sin (x)}{\cos (x)} - \frac{8}{\cos (x)} - 1 = - 8 \sec (x)$

Step 12

Simplify the right side.

Tap for more steps...

Step 12.1

Simplify $- 8 \sec (x)$ .

Tap for more steps...

Step 12.1.1

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

$\frac{\sin (x)}{\cos (x)} - \frac{8}{\cos (x)} - 1 = - 8 \frac{1}{\cos (x)}$

Step 12.1.2

Combine $- 8$ and $\frac{1}{\cos (x)}$ .

$\frac{\sin (x)}{\cos (x)} - \frac{8}{\cos (x)} - 1 = \frac{- 8}{\cos (x)}$

Step 12.1.3

Move the negative in front of the fraction.

$\frac{\sin (x)}{\cos (x)} - \frac{8}{\cos (x)} - 1 = - \frac{8}{\cos (x)}$

Step 13

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

$\cos (x) (\frac{\sin (x)}{\cos (x)} - \frac{8}{\cos (x)} - 1) = \cos (x) (- \frac{8}{\cos (x)})$

Step 14

Apply the distributive property.

$\cos (x) \frac{\sin (x)}{\cos (x)} + \cos (x) (- \frac{8}{\cos (x)}) + \cos (x) \cdot - 1 = \cos (x) (- \frac{8}{\cos (x)})$

Step 15

Simplify.

Tap for more steps...

Step 15.1

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

Tap for more steps...

Step 15.1.1

Cancel the common factor.

$\cos (x) \frac{\sin (x)}{\cos (x)} + \cos (x) (- \frac{8}{\cos (x)}) + \cos (x) \cdot - 1 = \cos (x) (- \frac{8}{\cos (x)})$

Step 15.1.2

Rewrite the expression.

$\sin (x) + \cos (x) (- \frac{8}{\cos (x)}) + \cos (x) \cdot - 1 = \cos (x) (- \frac{8}{\cos (x)})$

Step 15.2

Rewrite using the commutative property of multiplication.

$\sin (x) - \cos (x) \frac{8}{\cos (x)} + \cos (x) \cdot - 1 = \cos (x) (- \frac{8}{\cos (x)})$

Step 15.3

Move $- 1$ to the left of $\cos (x)$ .

$\sin (x) - \cos (x) \frac{8}{\cos (x)} - 1 \cdot \cos (x) = \cos (x) (- \frac{8}{\cos (x)})$

Step 16

Simplify each term.

Tap for more steps...

Step 16.1

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

Tap for more steps...

Step 16.1.1

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

$\sin (x) + \cos (x) \cdot - 1 \frac{8}{\cos (x)} - 1 \cdot \cos (x) = \cos (x) (- \frac{8}{\cos (x)})$

Step 16.1.2

Cancel the common factor.

$\sin (x) + \cos (x) \cdot - 1 \frac{8}{\cos (x)} - 1 \cdot \cos (x) = \cos (x) (- \frac{8}{\cos (x)})$

Step 16.1.3

Rewrite the expression.

$\sin (x) - 1 \cdot 8 - 1 \cdot \cos (x) = \cos (x) (- \frac{8}{\cos (x)})$

Step 16.2

Multiply $- 1$ by $8$ .

$\sin (x) - 8 - 1 \cdot \cos (x) = \cos (x) (- \frac{8}{\cos (x)})$

Step 16.3

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

$\sin (x) - 8 - \cos (x) = \cos (x) (- \frac{8}{\cos (x)})$

Step 17

Rewrite using the commutative property of multiplication.

$\sin (x) - 8 - \cos (x) = - \cos (x) \frac{8}{\cos (x)}$

Step 18

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

Tap for more steps...

Step 18.1

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

$\sin (x) - 8 - \cos (x) = \cos (x) \cdot - 1 \frac{8}{\cos (x)}$

Step 18.2

Cancel the common factor.

$\sin (x) - 8 - \cos (x) = \cos (x) \cdot - 1 \frac{8}{\cos (x)}$

Step 18.3

Rewrite the expression.

$\sin (x) - 8 - \cos (x) = - 1 \cdot 8$

Step 19

Multiply $- 1$ by $8$ .

$\sin (x) - 8 - \cos (x) = - 8$

Step 20

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

$\frac{\sin (x)}{\cos (x)} + \frac{- 8}{\cos (x)} + \frac{- \cos (x)}{\cos (x)} = \frac{- 8}{\cos (x)}$

Step 21

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

$\tan (x) + \frac{- 8}{\cos (x)} + \frac{- \cos (x)}{\cos (x)} = \frac{- 8}{\cos (x)}$

Step 22

Separate fractions.

$\tan (x) + \frac{- 8}{1} \cdot \frac{1}{\cos (x)} + \frac{- \cos (x)}{\cos (x)} = \frac{- 8}{\cos (x)}$

Step 23

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

$\tan (x) + \frac{- 8}{1} \cdot \sec (x) + \frac{- \cos (x)}{\cos (x)} = \frac{- 8}{\cos (x)}$

Step 24

Divide $- 8$ by $1$ .

$\tan (x) - 8 \sec (x) + \frac{- \cos (x)}{\cos (x)} = \frac{- 8}{\cos (x)}$

Step 25

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

Tap for more steps...

Step 25.1

Cancel the common factor.

$\tan (x) - 8 \sec (x) + \frac{- \cos (x)}{\cos (x)} = \frac{- 8}{\cos (x)}$

Step 25.2

Divide $- 1$ by $1$ .

$\tan (x) - 8 \sec (x) - 1 = \frac{- 8}{\cos (x)}$

Step 26

Separate fractions.

$\tan (x) - 8 \sec (x) - 1 = \frac{- 8}{1} \cdot \frac{1}{\cos (x)}$

Step 27

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

$\tan (x) - 8 \sec (x) - 1 = \frac{- 8}{1} \cdot \sec (x)$

Step 28

Divide $- 8$ by $1$ .

$\tan (x) - 8 \sec (x) - 1 = - 8 \sec (x)$

Step 29

Simplify the left side.

Tap for more steps...

Step 29.1

Simplify each term.

Tap for more steps...

Step 29.1.1

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

$\frac{\sin (x)}{\cos (x)} - 8 \sec (x) - 1 = - 8 \sec (x)$

Step 29.1.2

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

$\frac{\sin (x)}{\cos (x)} - 8 \frac{1}{\cos (x)} - 1 = - 8 \sec (x)$

Step 29.1.3

Combine $- 8$ and $\frac{1}{\cos (x)}$ .

$\frac{\sin (x)}{\cos (x)} + \frac{- 8}{\cos (x)} - 1 = - 8 \sec (x)$

Step 29.1.4

Move the negative in front of the fraction.

$\frac{\sin (x)}{\cos (x)} - \frac{8}{\cos (x)} - 1 = - 8 \sec (x)$

Step 30

Simplify the right side.

Tap for more steps...

Step 30.1

Simplify $- 8 \sec (x)$ .

Tap for more steps...

Step 30.1.1

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

$\frac{\sin (x)}{\cos (x)} - \frac{8}{\cos (x)} - 1 = - 8 \frac{1}{\cos (x)}$

Step 30.1.2

Combine $- 8$ and $\frac{1}{\cos (x)}$ .

$\frac{\sin (x)}{\cos (x)} - \frac{8}{\cos (x)} - 1 = \frac{- 8}{\cos (x)}$

Step 30.1.3

Move the negative in front of the fraction.

$\frac{\sin (x)}{\cos (x)} - \frac{8}{\cos (x)} - 1 = - \frac{8}{\cos (x)}$

Step 31

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

$\cos (x) (\frac{\sin (x)}{\cos (x)} - \frac{8}{\cos (x)} - 1) = \cos (x) (- \frac{8}{\cos (x)})$

Step 32

Apply the distributive property.

$\cos (x) \frac{\sin (x)}{\cos (x)} + \cos (x) (- \frac{8}{\cos (x)}) + \cos (x) \cdot - 1 = \cos (x) (- \frac{8}{\cos (x)})$

Step 33

Simplify.

Tap for more steps...

Step 33.1

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

Tap for more steps...

Step 33.1.1

Cancel the common factor.

$\cos (x) \frac{\sin (x)}{\cos (x)} + \cos (x) (- \frac{8}{\cos (x)}) + \cos (x) \cdot - 1 = \cos (x) (- \frac{8}{\cos (x)})$

Step 33.1.2

Rewrite the expression.

$\sin (x) + \cos (x) (- \frac{8}{\cos (x)}) + \cos (x) \cdot - 1 = \cos (x) (- \frac{8}{\cos (x)})$

Step 33.2

Rewrite using the commutative property of multiplication.

$\sin (x) - \cos (x) \frac{8}{\cos (x)} + \cos (x) \cdot - 1 = \cos (x) (- \frac{8}{\cos (x)})$

Step 33.3

Move $- 1$ to the left of $\cos (x)$ .

$\sin (x) - \cos (x) \frac{8}{\cos (x)} - 1 \cdot \cos (x) = \cos (x) (- \frac{8}{\cos (x)})$

Step 34

Simplify each term.

Tap for more steps...

Step 34.1

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

Tap for more steps...

Step 34.1.1

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

$\sin (x) + \cos (x) \cdot - 1 \frac{8}{\cos (x)} - 1 \cdot \cos (x) = \cos (x) (- \frac{8}{\cos (x)})$

Step 34.1.2

Cancel the common factor.

$\sin (x) + \cos (x) \cdot - 1 \frac{8}{\cos (x)} - 1 \cdot \cos (x) = \cos (x) (- \frac{8}{\cos (x)})$

Step 34.1.3

Rewrite the expression.

$\sin (x) - 1 \cdot 8 - 1 \cdot \cos (x) = \cos (x) (- \frac{8}{\cos (x)})$

Step 34.2

Multiply $- 1$ by $8$ .

$\sin (x) - 8 - 1 \cdot \cos (x) = \cos (x) (- \frac{8}{\cos (x)})$

Step 34.3

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

$\sin (x) - 8 - \cos (x) = \cos (x) (- \frac{8}{\cos (x)})$

Step 35

Rewrite using the commutative property of multiplication.

$\sin (x) - 8 - \cos (x) = - \cos (x) \frac{8}{\cos (x)}$

Step 36

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

Tap for more steps...

Step 36.1

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

$\sin (x) - 8 - \cos (x) = \cos (x) \cdot - 1 \frac{8}{\cos (x)}$

Step 36.2

Cancel the common factor.

$\sin (x) - 8 - \cos (x) = \cos (x) \cdot - 1 \frac{8}{\cos (x)}$

Step 36.3

Rewrite the expression.

$\sin (x) - 8 - \cos (x) = - 1 \cdot 8$

Step 37

Multiply $- 1$ by $8$ .

$\sin (x) - 8 - \cos (x) = - 8$

Step 38

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

$\frac{\sin (x)}{\cos (x)} + \frac{- 8}{\cos (x)} + \frac{- \cos (x)}{\cos (x)} = \frac{- 8}{\cos (x)}$

Step 39

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

$\tan (x) + \frac{- 8}{\cos (x)} + \frac{- \cos (x)}{\cos (x)} = \frac{- 8}{\cos (x)}$

Step 40

Separate fractions.

$\tan (x) + \frac{- 8}{1} \cdot \frac{1}{\cos (x)} + \frac{- \cos (x)}{\cos (x)} = \frac{- 8}{\cos (x)}$

Step 41

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

$\tan (x) + \frac{- 8}{1} \cdot \sec (x) + \frac{- \cos (x)}{\cos (x)} = \frac{- 8}{\cos (x)}$

Step 42

Divide $- 8$ by $1$ .

$\tan (x) - 8 \sec (x) + \frac{- \cos (x)}{\cos (x)} = \frac{- 8}{\cos (x)}$

Step 43

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

Tap for more steps...

Step 43.1

Cancel the common factor.

$\tan (x) - 8 \sec (x) + \frac{- \cos (x)}{\cos (x)} = \frac{- 8}{\cos (x)}$

Step 43.2

Divide $- 1$ by $1$ .

$\tan (x) - 8 \sec (x) - 1 = \frac{- 8}{\cos (x)}$

Step 44

Separate fractions.

$\tan (x) - 8 \sec (x) - 1 = \frac{- 8}{1} \cdot \frac{1}{\cos (x)}$

Step 45

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

$\tan (x) - 8 \sec (x) - 1 = \frac{- 8}{1} \cdot \sec (x)$

Step 46

Divide $- 8$ by $1$ .

$\tan (x) - 8 \sec (x) - 1 = - 8 \sec (x)$

Step 47

Simplify the left side.

Tap for more steps...

Step 47.1

Simplify each term.

Tap for more steps...

Step 47.1.1

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

$\frac{\sin (x)}{\cos (x)} - 8 \sec (x) - 1 = - 8 \sec (x)$

Step 47.1.2

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

$\frac{\sin (x)}{\cos (x)} - 8 \frac{1}{\cos (x)} - 1 = - 8 \sec (x)$

Step 47.1.3

Combine $- 8$ and $\frac{1}{\cos (x)}$ .

$\frac{\sin (x)}{\cos (x)} + \frac{- 8}{\cos (x)} - 1 = - 8 \sec (x)$

Step 47.1.4

Move the negative in front of the fraction.

$\frac{\sin (x)}{\cos (x)} - \frac{8}{\cos (x)} - 1 = - 8 \sec (x)$

Step 48

Simplify the right side.

Tap for more steps...

Step 48.1

Simplify $- 8 \sec (x)$ .

Tap for more steps...

Step 48.1.1

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

$\frac{\sin (x)}{\cos (x)} - \frac{8}{\cos (x)} - 1 = - 8 \frac{1}{\cos (x)}$

Step 48.1.2

Combine $- 8$ and $\frac{1}{\cos (x)}$ .

$\frac{\sin (x)}{\cos (x)} - \frac{8}{\cos (x)} - 1 = \frac{- 8}{\cos (x)}$

Step 48.1.3

Move the negative in front of the fraction.

$\frac{\sin (x)}{\cos (x)} - \frac{8}{\cos (x)} - 1 = - \frac{8}{\cos (x)}$

Step 49

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

$\cos (x) (\frac{\sin (x)}{\cos (x)} - \frac{8}{\cos (x)} - 1) = \cos (x) (- \frac{8}{\cos (x)})$

Step 50

Apply the distributive property.

$\cos (x) \frac{\sin (x)}{\cos (x)} + \cos (x) (- \frac{8}{\cos (x)}) + \cos (x) \cdot - 1 = \cos (x) (- \frac{8}{\cos (x)})$

Step 51

Simplify.

Tap for more steps...

Step 51.1

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

Tap for more steps...

Step 51.1.1

Cancel the common factor.

$\cos (x) \frac{\sin (x)}{\cos (x)} + \cos (x) (- \frac{8}{\cos (x)}) + \cos (x) \cdot - 1 = \cos (x) (- \frac{8}{\cos (x)})$

Step 51.1.2

Rewrite the expression.

$\sin (x) + \cos (x) (- \frac{8}{\cos (x)}) + \cos (x) \cdot - 1 = \cos (x) (- \frac{8}{\cos (x)})$

Step 51.2

Rewrite using the commutative property of multiplication.

$\sin (x) - \cos (x) \frac{8}{\cos (x)} + \cos (x) \cdot - 1 = \cos (x) (- \frac{8}{\cos (x)})$

Step 51.3

Move $- 1$ to the left of $\cos (x)$ .

$\sin (x) - \cos (x) \frac{8}{\cos (x)} - 1 \cdot \cos (x) = \cos (x) (- \frac{8}{\cos (x)})$

Step 52

Simplify each term.

Tap for more steps...

Step 52.1

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

Tap for more steps...

Step 52.1.1

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

$\sin (x) + \cos (x) \cdot - 1 \frac{8}{\cos (x)} - 1 \cdot \cos (x) = \cos (x) (- \frac{8}{\cos (x)})$

Step 52.1.2

Cancel the common factor.

$\sin (x) + \cos (x) \cdot - 1 \frac{8}{\cos (x)} - 1 \cdot \cos (x) = \cos (x) (- \frac{8}{\cos (x)})$

Step 52.1.3

Rewrite the expression.

$\sin (x) - 1 \cdot 8 - 1 \cdot \cos (x) = \cos (x) (- \frac{8}{\cos (x)})$

Step 52.2

Multiply $- 1$ by $8$ .

$\sin (x) - 8 - 1 \cdot \cos (x) = \cos (x) (- \frac{8}{\cos (x)})$

Step 52.3

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

$\sin (x) - 8 - \cos (x) = \cos (x) (- \frac{8}{\cos (x)})$

Step 53

Rewrite using the commutative property of multiplication.

$\sin (x) - 8 - \cos (x) = - \cos (x) \frac{8}{\cos (x)}$

Step 54

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

Tap for more steps...

Step 54.1

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

$\sin (x) - 8 - \cos (x) = \cos (x) \cdot - 1 \frac{8}{\cos (x)}$

Step 54.2

Cancel the common factor.

$\sin (x) - 8 - \cos (x) = \cos (x) \cdot - 1 \frac{8}{\cos (x)}$

Step 54.3

Rewrite the expression.

$\sin (x) - 8 - \cos (x) = - 1 \cdot 8$

Step 55

Multiply $- 1$ by $8$ .

$\sin (x) - 8 - \cos (x) = - 8$

Step 56

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

$\frac{\sin (x)}{\cos (x)} + \frac{- 8}{\cos (x)} + \frac{- \cos (x)}{\cos (x)} = \frac{- 8}{\cos (x)}$

Step 57

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

$\tan (x) + \frac{- 8}{\cos (x)} + \frac{- \cos (x)}{\cos (x)} = \frac{- 8}{\cos (x)}$

Step 58

Separate fractions.

$\tan (x) + \frac{- 8}{1} \cdot \frac{1}{\cos (x)} + \frac{- \cos (x)}{\cos (x)} = \frac{- 8}{\cos (x)}$

Step 59

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

$\tan (x) + \frac{- 8}{1} \cdot \sec (x) + \frac{- \cos (x)}{\cos (x)} = \frac{- 8}{\cos (x)}$

Step 60

Divide $- 8$ by $1$ .

$\tan (x) - 8 \sec (x) + \frac{- \cos (x)}{\cos (x)} = \frac{- 8}{\cos (x)}$

Step 61

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

Tap for more steps...

Step 61.1

Cancel the common factor.

$\tan (x) - 8 \sec (x) + \frac{- \cos (x)}{\cos (x)} = \frac{- 8}{\cos (x)}$

Step 61.2

Divide $- 1$ by $1$ .

$\tan (x) - 8 \sec (x) - 1 = \frac{- 8}{\cos (x)}$

Step 62

Separate fractions.

$\tan (x) - 8 \sec (x) - 1 = \frac{- 8}{1} \cdot \frac{1}{\cos (x)}$

Step 63

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

$\tan (x) - 8 \sec (x) - 1 = \frac{- 8}{1} \cdot \sec (x)$

Step 64

Divide $- 8$ by $1$ .

$\tan (x) - 8 \sec (x) - 1 = - 8 \sec (x)$

Step 65

Move all terms containing $x$ to the left side of the equation.

Tap for more steps...

Step 65.1

Add $8 \sec (x)$ to both sides of the equation.

$\tan (x) - 8 \sec (x) - 1 + 8 \sec (x) = 0$

Step 65.2

Combine the opposite terms in $\tan (x) - 8 \sec (x) - 1 + 8 \sec (x)$ .

Tap for more steps...

Step 65.2.1

Add $- 8 \sec (x)$ and $8 \sec (x)$ .

$\tan (x) + 0 - 1 = 0$

Step 65.2.2

Add $\tan (x)$ and $0$ .

$\tan (x) - 1 = 0$

Step 66

Add $1$ to both sides of the equation.

$\tan (x) = 1$

Step 67

Take the inverse tangent of both sides of the equation to extract $x$ from inside the tangent.

$x = \arctan (1)$

Step 68

Simplify the right side.

Tap for more steps...

Step 68.1

The exact value of $\arctan (1)$ is $\frac{π}{4}$ .

$x = \frac{π}{4}$

Step 69

The tangent function is positive in the first and third quadrants. To find the second solution, add the reference angle from $π$ to find the solution in the fourth quadrant.

$x = π + \frac{π}{4}$

Step 70

Simplify $π + \frac{π}{4}$ .

Tap for more steps...

Step 70.1

To write $π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = π \cdot \frac{4}{4} + \frac{π}{4}$

Step 70.2

Combine fractions.

Tap for more steps...

Step 70.2.1

Combine $π$ and $\frac{4}{4}$ .

$x = \frac{π \cdot 4}{4} + \frac{π}{4}$

Step 70.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 4 + π}{4}$

Step 70.3

Simplify the numerator.

Tap for more steps...

Step 70.3.1

Move $4$ to the left of $π$ .

$x = \frac{4 \cdot π + π}{4}$

Step 70.3.2

Add $4 π$ and $π$ .

$x = \frac{5 π}{4}$

Step 71

Find the period of $\tan (x)$ .

Tap for more steps...

Step 71.1

The period of the function can be calculated using $\frac{π}{| b |}$ .

$\frac{π}{| b |}$

Step 71.2

Replace $b$ with $1$ in the formula for period.

$\frac{π}{| 1 |}$

Step 71.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{π}{1}$

Step 71.4

Divide $π$ by $1$ .

$π$

Step 72

The period of the $\tan (x)$ function is $π$ so values will repeat every $π$ radians in both directions.

$x = \frac{π}{4} + π n, \frac{5 π}{4} + π n$ , for any integer $n$

Step 73

Consolidate the answers.

$x = \frac{π}{4} + π n$ , for any integer $n$