Trigonometry Examples

$\frac{1 + \tan (x)}{1 + \cot (x)} = \sec^{2} (x)$

Step 1

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

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Step 1.1

Subtract $\sec^{2} (x)$ from both sides of the equation.

$\frac{1 + \tan (x)}{1 + \cot (x)} - \sec^{2} (x) = 0$

Step 1.2

To write $- \sec^{2} (x)$ as a fraction with a common denominator, multiply by $\frac{1 + \cot (x)}{1 + \cot (x)}$ .

$\frac{1 + \tan (x)}{1 + \cot (x)} - \sec^{2} (x) \cdot \frac{1 + \cot (x)}{1 + \cot (x)} = 0$

Step 1.3

Combine $- \sec^{2} (x)$ and $\frac{1 + \cot (x)}{1 + \cot (x)}$ .

$\frac{1 + \tan (x)}{1 + \cot (x)} + \frac{- \sec^{2} (x) (1 + \cot (x))}{1 + \cot (x)} = 0$

Step 1.4

Combine the numerators over the common denominator.

$\frac{1 + \tan (x) - \sec^{2} (x) (1 + \cot (x))}{1 + \cot (x)} = 0$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Apply the distributive property.

$\frac{1 + \tan (x) - \sec^{2} (x) \cdot 1 - \sec^{2} (x) \cot (x)}{1 + \cot (x)} = 0$

Step 1.5.2

Multiply $- 1$ by $1$ .

$\frac{1 + \tan (x) - \sec^{2} (x) - \sec^{2} (x) \cot (x)}{1 + \cot (x)} = 0$

Step 1.5.3

Move $- \sec^{2} (x)$ .

$\frac{1 - \sec^{2} (x) + \tan (x) - \sec^{2} (x) \cot (x)}{1 + \cot (x)} = 0$

Step 1.5.4

Reorder $1$ and $- \sec^{2} (x)$ .

$\frac{- \sec^{2} (x) + 1 + \tan (x) - \sec^{2} (x) \cot (x)}{1 + \cot (x)} = 0$

Step 1.5.5

Factor $- 1$ out of $- \sec^{2} (x)$ .

$\frac{- (\sec^{2} (x)) + 1 + \tan (x) - \sec^{2} (x) \cot (x)}{1 + \cot (x)} = 0$

Step 1.5.6

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- \sec^{2} (x) - 1 \cdot - 1 + \tan (x) - \sec^{2} (x) \cot (x)}{1 + \cot (x)} = 0$

Step 1.5.7

Factor $- 1$ out of $- \sec^{2} (x) - 1 \cdot - 1$ .

$\frac{- (\sec^{2} (x) - 1) + \tan (x) - \sec^{2} (x) \cot (x)}{1 + \cot (x)} = 0$

Step 1.5.8

Apply pythagorean identity.

$\frac{- \tan^{2} (x) + \tan (x) - \sec^{2} (x) \cot (x)}{1 + \cot (x)} = 0$

Step 1.6

Factor $- 1$ out of $- \tan^{2} (x)$ .

$\frac{- (\tan^{2} (x)) + \tan (x) - \sec^{2} (x) \cot (x)}{1 + \cot (x)} = 0$

Step 1.7

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

$\frac{- (\tan^{2} (x)) - 1 (- \tan (x)) - \sec^{2} (x) \cot (x)}{1 + \cot (x)} = 0$

Step 1.8

Factor $- 1$ out of $- (\tan^{2} (x)) - 1 (- \tan (x))$ .

$\frac{- (\tan^{2} (x) - \tan (x)) - \sec^{2} (x) \cot (x)}{1 + \cot (x)} = 0$

Step 1.9

Factor $- 1$ out of $- \sec^{2} (x) \cot (x)$ .

$\frac{- (\tan^{2} (x) - \tan (x)) - (\sec^{2} (x) \cot (x))}{1 + \cot (x)} = 0$

Step 1.10

Factor $- 1$ out of $- (\tan^{2} (x) - \tan (x)) - (\sec^{2} (x) \cot (x))$ .

$\frac{- (\tan^{2} (x) - \tan (x) + \sec^{2} (x) \cot (x))}{1 + \cot (x)} = 0$

Step 1.11

Rewrite $- (\tan^{2} (x) - \tan (x) + \sec^{2} (x) \cot (x))$ as $- 1 (\tan^{2} (x) - \tan (x) + \sec^{2} (x) \cot (x))$ .

$\frac{- 1 (\tan^{2} (x) - \tan (x) + \sec^{2} (x) \cot (x))}{1 + \cot (x)} = 0$

Step 1.12

Move the negative in front of the fraction.

$- \frac{\tan^{2} (x) - \tan (x) + \sec^{2} (x) \cot (x)}{1 + \cot (x)} = 0$

Step 2

Set the numerator equal to zero.

$\tan^{2} (x) - \tan (x) + \sec^{2} (x) \cot (x) = 0$

Step 3

Solve the equation for $x$ .

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify each term.

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Step 3.1.1.1

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

${(\frac{\sin (x)}{\cos (x)})}^{2} - \tan (x) + \sec^{2} (x) \cot (x) = 0$

Step 3.1.1.2

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

$\frac{\sin^{2} (x)}{\cos^{2} (x)} - \tan (x) + \sec^{2} (x) \cot (x) = 0$

Step 3.1.1.3

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

$\frac{\sin^{2} (x)}{\cos^{2} (x)} - \frac{\sin (x)}{\cos (x)} + \sec^{2} (x) \cot (x) = 0$

Step 3.1.1.4

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

$\frac{\sin^{2} (x)}{\cos^{2} (x)} - \frac{\sin (x)}{\cos (x)} + {(\frac{1}{\cos (x)})}^{2} \cot (x) = 0$

Step 3.1.1.5

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

$\frac{\sin^{2} (x)}{\cos^{2} (x)} - \frac{\sin (x)}{\cos (x)} + \frac{1^{2}}{\cos^{2} (x)} \cot (x) = 0$

Step 3.1.1.6

One to any power is one.

$\frac{\sin^{2} (x)}{\cos^{2} (x)} - \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos^{2} (x)} \cot (x) = 0$

Step 3.1.1.7

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

$\frac{\sin^{2} (x)}{\cos^{2} (x)} - \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos^{2} (x)} \cdot \frac{\cos (x)}{\sin (x)} = 0$

Step 3.1.1.8

Combine.

$\frac{\sin^{2} (x)}{\cos^{2} (x)} - \frac{\sin (x)}{\cos (x)} + \frac{1 \cos (x)}{\cos^{2} (x) \sin (x)} = 0$

Step 3.1.1.9

Cancel the common factor of $\cos (x)$ and $\cos^{2} (x)$ .

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Step 3.1.1.9.1

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

$\frac{\sin^{2} (x)}{\cos^{2} (x)} - \frac{\sin (x)}{\cos (x)} + \frac{\cos (x) \cdot 1}{\cos^{2} (x) \sin (x)} = 0$

Step 3.1.1.9.2

Cancel the common factors.

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Step 3.1.1.9.2.1

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

$\frac{\sin^{2} (x)}{\cos^{2} (x)} - \frac{\sin (x)}{\cos (x)} + \frac{\cos (x) \cdot 1}{\cos (x) (\cos (x) \sin (x))} = 0$

Step 3.1.1.9.2.2

Cancel the common factor.

$\frac{\sin^{2} (x)}{\cos^{2} (x)} - \frac{\sin (x)}{\cos (x)} + \frac{\cos (x) \cdot 1}{\cos (x) (\cos (x) \sin (x))} = 0$

Step 3.1.1.9.2.3

Rewrite the expression.

$\frac{\sin^{2} (x)}{\cos^{2} (x)} - \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x) \sin (x)} = 0$

Step 3.2

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

$\cos (x) (\frac{\sin^{2} (x)}{\cos^{2} (x)} - \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x) \sin (x)}) = \cos (x) \cdot 0$

Step 3.3

Apply the distributive property.

$\cos (x) \frac{\sin^{2} (x)}{\cos^{2} (x)} + \cos (x) (- \frac{\sin (x)}{\cos (x)}) + \cos (x) \frac{1}{\cos (x) \sin (x)} = \cos (x) \cdot 0$

Step 3.4

Simplify.

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Step 3.4.1

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

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Step 3.4.1.1

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

$\cos (x) \frac{\sin^{2} (x)}{\cos (x) \cos (x)} + \cos (x) (- \frac{\sin (x)}{\cos (x)}) + \cos (x) \frac{1}{\cos (x) \sin (x)} = \cos (x) \cdot 0$

Step 3.4.1.2

Cancel the common factor.

$\cos (x) \frac{\sin^{2} (x)}{\cos (x) \cos (x)} + \cos (x) (- \frac{\sin (x)}{\cos (x)}) + \cos (x) \frac{1}{\cos (x) \sin (x)} = \cos (x) \cdot 0$

Step 3.4.1.3

Rewrite the expression.

$\frac{\sin^{2} (x)}{\cos (x)} + \cos (x) (- \frac{\sin (x)}{\cos (x)}) + \cos (x) \frac{1}{\cos (x) \sin (x)} = \cos (x) \cdot 0$

Step 3.4.2

Rewrite using the commutative property of multiplication.

$\frac{\sin^{2} (x)}{\cos (x)} - \cos (x) \frac{\sin (x)}{\cos (x)} + \cos (x) \frac{1}{\cos (x) \sin (x)} = \cos (x) \cdot 0$

Step 3.4.3

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

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Step 3.4.3.1

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

$\frac{\sin^{2} (x)}{\cos (x)} - \cos (x) \frac{\sin (x)}{\cos (x)} + \cos (x) \frac{1}{\cos (x) (\sin (x))} = \cos (x) \cdot 0$

Step 3.4.3.2

Cancel the common factor.

$\frac{\sin^{2} (x)}{\cos (x)} - \cos (x) \frac{\sin (x)}{\cos (x)} + \cos (x) \frac{1}{\cos (x) \sin (x)} = \cos (x) \cdot 0$

Step 3.4.3.3

Rewrite the expression.

$\frac{\sin^{2} (x)}{\cos (x)} - \cos (x) \frac{\sin (x)}{\cos (x)} + \frac{1}{\sin (x)} = \cos (x) \cdot 0$

Step 3.5

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

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Step 3.5.1

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

$\frac{\sin^{2} (x)}{\cos (x)} + \cos (x) \cdot - 1 \frac{\sin (x)}{\cos (x)} + \frac{1}{\sin (x)} = \cos (x) \cdot 0$

Step 3.5.2

Cancel the common factor.

$\frac{\sin^{2} (x)}{\cos (x)} + \cos (x) \cdot - 1 \frac{\sin (x)}{\cos (x)} + \frac{1}{\sin (x)} = \cos (x) \cdot 0$

Step 3.5.3

Rewrite the expression.

$\frac{\sin^{2} (x)}{\cos (x)} - \sin (x) + \frac{1}{\sin (x)} = \cos (x) \cdot 0$

Step 3.6

Multiply $\cos (x)$ by $0$ .

$\frac{\sin^{2} (x)}{\cos (x)} - \sin (x) + \frac{1}{\sin (x)} = 0$

Step 3.7

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

$\sin (x) (\frac{\sin^{2} (x)}{\cos (x)} - \sin (x) + \frac{1}{\sin (x)}) = \sin (x) \cdot 0$

Step 3.8

Apply the distributive property.

$\sin (x) \frac{\sin^{2} (x)}{\cos (x)} + \sin (x) (- \sin (x)) + \sin (x) \frac{1}{\sin (x)} = \sin (x) \cdot 0$

Step 3.9

Simplify.

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Step 3.9.1

Multiply $\sin (x) \frac{\sin^{2} (x)}{\cos (x)}$ .

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Step 3.9.1.1

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

$\frac{\sin (x) \sin^{2} (x)}{\cos (x)} + \sin (x) (- \sin (x)) + \sin (x) \frac{1}{\sin (x)} = \sin (x) \cdot 0$

Step 3.9.1.2

Multiply $\sin (x)$ by $\sin^{2} (x)$ by adding the exponents.

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Step 3.9.1.2.1

Multiply $\sin (x)$ by $\sin^{2} (x)$ .

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Step 3.9.1.2.1.1

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

$\frac{\sin^{1} (x) \sin^{2} (x)}{\cos (x)} + \sin (x) (- \sin (x)) + \sin (x) \frac{1}{\sin (x)} = \sin (x) \cdot 0$

Step 3.9.1.2.1.2

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

$\frac{{\sin (x)}^{1 + 2}}{\cos (x)} + \sin (x) (- \sin (x)) + \sin (x) \frac{1}{\sin (x)} = \sin (x) \cdot 0$

Step 3.9.1.2.2

Add $1$ and $2$ .

$\frac{\sin^{3} (x)}{\cos (x)} + \sin (x) (- \sin (x)) + \sin (x) \frac{1}{\sin (x)} = \sin (x) \cdot 0$

Step 3.9.2

Rewrite using the commutative property of multiplication.

$\frac{\sin^{3} (x)}{\cos (x)} - \sin (x) \sin (x) + \sin (x) \frac{1}{\sin (x)} = \sin (x) \cdot 0$

Step 3.9.3

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

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Step 3.9.3.1

Cancel the common factor.

$\frac{\sin^{3} (x)}{\cos (x)} - \sin (x) \sin (x) + \sin (x) \frac{1}{\sin (x)} = \sin (x) \cdot 0$

Step 3.9.3.2

Rewrite the expression.

$\frac{\sin^{3} (x)}{\cos (x)} - \sin (x) \sin (x) + 1 = \sin (x) \cdot 0$

Step 3.10

Multiply $- \sin (x) \sin (x)$ .

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Step 3.10.1

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

$\frac{\sin^{3} (x)}{\cos (x)} - (\sin^{1} (x) \sin (x)) + 1 = \sin (x) \cdot 0$

Step 3.10.2

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

$\frac{\sin^{3} (x)}{\cos (x)} - (\sin^{1} (x) \sin^{1} (x)) + 1 = \sin (x) \cdot 0$

Step 3.10.3

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

$\frac{\sin^{3} (x)}{\cos (x)} - {\sin (x)}^{1 + 1} + 1 = \sin (x) \cdot 0$

Step 3.10.4

Add $1$ and $1$ .

$\frac{\sin^{3} (x)}{\cos (x)} - \sin^{2} (x) + 1 = \sin (x) \cdot 0$

Step 3.11

Reorder $- \sin^{2} (x)$ and $1$ .

$\frac{\sin^{3} (x)}{\cos (x)} + 1 - \sin^{2} (x) = \sin (x) \cdot 0$

Step 3.12

Apply pythagorean identity.

$\frac{\sin^{3} (x)}{\cos (x)} + \cos^{2} (x) = \sin (x) \cdot 0$

Step 3.13

Multiply $\sin (x)$ by $0$ .

$\frac{\sin^{3} (x)}{\cos (x)} + \cos^{2} (x) = 0$

Step 3.14

Replace $\cos^{2} (x)$ with $1 - \sin^{2} (x)$ .

$\frac{\sin^{3} (x)}{\cos (x) (1 - \sin^{2} (x))} = 0$

Step 3.15

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

$\frac{\frac{\sin^{3} (x)}{\cos (x) (1 - \sin^{2} (x))}}{\cos (x)} = \frac{0}{\cos (x)}$

Step 3.16

Apply pythagorean identity.

$\frac{\frac{\sin^{3} (x)}{\cos (x) \cos^{2} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Step 3.17

Separate fractions.

$\frac{\frac{\sin^{3} (x)}{\cos (x) \cos^{2} (x)}}{1} \cdot \frac{1}{\cos (x)} = \frac{0}{\cos (x)}$

Step 3.18

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

$\frac{\frac{\sin^{3} (x)}{\cos (x) \cos^{2} (x)}}{1} \cdot \sec (x) = \frac{0}{\cos (x)}$

Step 3.19

Divide $\frac{\sin^{3} (x)}{\cos (x) \cos^{2} (x)}$ by $1$ .

$\frac{\sin^{3} (x)}{\cos (x) \cos^{2} (x)} \cdot \sec (x) = \frac{0}{\cos (x)}$

Step 3.20

Multiply $\cos (x)$ by $\cos^{2} (x)$ by adding the exponents.

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Step 3.20.1

Multiply $\cos (x)$ by $\cos^{2} (x)$ .

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Step 3.20.1.1

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

$\frac{\sin^{3} (x)}{\cos (x) \cos^{2} (x)} \cdot \sec (x) = \frac{0}{\cos (x)}$

Step 3.20.1.2

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

$\frac{\sin^{3} (x)}{{\cos (x)}^{1 + 2}} \cdot \sec (x) = \frac{0}{\cos (x)}$

Step 3.20.2

Add $1$ and $2$ .

$\frac{\sin^{3} (x)}{\cos^{3} (x)} \cdot \sec (x) = \frac{0}{\cos (x)}$

Step 3.21

Convert from $\frac{\sin^{3} (x)}{\cos^{3} (x)}$ to $\tan^{3} (x)$ .

$\tan^{3} (x) \sec (x) = \frac{0}{\cos (x)}$

Step 3.22

Separate fractions.

$\tan^{3} (x) \sec (x) = \frac{0}{1} \cdot \frac{1}{\cos (x)}$

Step 3.23

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

$\tan^{3} (x) \sec (x) = \frac{0}{1} \cdot \sec (x)$

Step 3.24

Divide $0$ by $1$ .

$\tan^{3} (x) \sec (x) = 0 \sec (x)$

Step 3.25

Multiply $0$ by $\sec (x)$ .

$\tan^{3} (x) \sec (x) = 0$

Step 3.26

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$\tan^{3} (x) = 0$

$\sec (x) = 0$

Step 3.27

Set $\tan^{3} (x)$ equal to $0$ and solve for $x$ .

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Step 3.27.1

Set $\tan^{3} (x)$ equal to $0$ .

$\tan^{3} (x) = 0$

Step 3.27.2

Solve $\tan^{3} (x) = 0$ for $x$ .

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Step 3.27.2.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$\tan (x) = \sqrt[3]{0}$

Step 3.27.2.2

Simplify $\sqrt[3]{0}$ .

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Step 3.27.2.2.1

Rewrite $0$ as $0^{3}$ .

$\tan (x) = \sqrt[3]{0^{3}}$

Step 3.27.2.2.2

Pull terms out from under the radical, assuming real numbers.

$\tan (x) = 0$

Step 3.27.2.3

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

$x = \arctan (0)$

Step 3.27.2.4

Simplify the right side.

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Step 3.27.2.4.1

The exact value of $\arctan (0)$ is $0$ .

$x = 0$

Step 3.27.2.5

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.

$x = π + 0$

Step 3.27.2.6

Add $π$ and $0$ .

$x = π$

Step 3.27.2.7

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

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Step 3.27.2.7.1

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

$\frac{π}{| b |}$

Step 3.27.2.7.2

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

$\frac{π}{| 1 |}$

Step 3.27.2.7.3

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

$\frac{π}{1}$

Step 3.27.2.7.4

Divide $π$ by $1$ .

$π$

Step 3.27.2.8

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

$x = π n, π + π n$ , for any integer $n$

Step 3.28

Set $\sec (x)$ equal to $0$ and solve for $x$ .

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Step 3.28.1

Set $\sec (x)$ equal to $0$ .

$\sec (x) = 0$

Step 3.28.2

The range of secant is $y \leq - 1$ and $y \geq 1$ . Since $0$ does not fall in this range, there is no solution.

No solution

Step 3.29

The final solution is all the values that make $\tan^{3} (x) \sec (x) = 0$ true.

$x = π n, π + π n$ , for any integer $n$

Step 4

Consolidate the answers.

$x = π n$ , for any integer $n$

Step 5

Exclude the solutions that do not make $\frac{1 + \tan (x)}{1 + \cot (x)} = \sec^{2} (x)$ true.

No solution