Trigonometry Examples

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

Step 1

Start on the left side.

$\frac{\cos (x) \cdot \cot (x)}{\cot (x) - \cos (x)}$

Step 2

Convert to sines and cosines.

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Step 2.1

Write $\cot (x)$ in sines and cosines using the quotient identity.

$\frac{\cos (x) \cdot \frac{\cos (x)}{\sin (x)}}{\cot (x) - \cos (x)}$

Step 2.2

Write $\cot (x)$ in sines and cosines using the quotient identity.

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

Step 3

Write $\cos (x)$ as a fraction with denominator $1$ .

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

Step 4

Combine.

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

Step 5

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

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

Step 6

Multiply $\sin (x)$ by $1$ .

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

Step 7

Apply Pythagorean identity in reverse.

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

Step 8

Simplify.

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Step 8.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 8.2

Simplify the numerator.

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Step 8.2.1

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

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

Step 8.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = \sin (x)$ .

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

Step 8.3

Simplify the denominator.

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Step 8.3.1

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

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Step 8.3.1.1

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

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

Step 8.3.1.2

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

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

Step 8.3.1.3

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

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

Step 8.3.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{\sin (x)}{\sin (x)}$ .

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

Step 8.3.3

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

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

Step 8.3.4

Combine the numerators over the common denominator.

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

Step 8.4

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

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

Step 8.5

Multiply the numerator by the reciprocal of the denominator.

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

Step 8.6

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

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

Step 8.7

Cancel the common factor of $1 - \sin (x)$ .

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Step 8.7.1

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

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

Step 8.7.2

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

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

Step 8.7.3

Cancel the common factor.

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

Step 8.7.4

Rewrite the expression.

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

Step 8.8

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

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Step 8.8.1

Cancel the common factor.

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

Step 8.8.2

Rewrite the expression.

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

Step 8.9

Apply the distributive property.

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

Step 8.10

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

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

Step 8.11

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

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

Step 9

Combine the numerators over the common denominator.

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

Step 10

Rewrite $\frac{1 + \sin (x)}{\cos (x)}$ as $\frac{\cot (x) + \cos (x)}{\cos (x) \cdot \cot (x)}$ .

$\frac{\cot (x) + \cos (x)}{\cos (x) \cdot \cot (x)}$

Step 11

Because the two sides have been shown to be equivalent, the equation is an identity.

$\frac{\cos (x) \cdot \cot (x)}{\cot (x) - \cos (x)} = \frac{\cot (x) + \cos (x)}{\cos (x) \cdot \cot (x)}$ is an identity