Trigonometry Examples

$\frac{\csc (x) \cot (x)}{\sec (x) \tan (x)} = \cot^{3} (x)$

Step 1

Start on the left side.

$\frac{\csc (x) \cot (x)}{\sec (x) \tan (x)}$

Step 2

Convert to sines and cosines.

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Step 2.1

Apply the reciprocal identity to $\csc (x)$ .

$\frac{\frac{1}{\sin (x)} \cot (x)}{\sec (x) \tan (x)}$

Step 2.2

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

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

Step 2.3

Apply the reciprocal identity to $\sec (x)$ .

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

Step 2.4

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

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

Step 3

Simplify.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Simplify the denominator.

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.3.4

Add $1$ and $1$ .

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

Step 3.4

Simplify the denominator.

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Step 3.4.1

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

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

Step 3.4.2

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

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

Step 3.4.3

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

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

Step 3.4.4

Add $1$ and $1$ .

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

Step 3.5

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.6

Combine.

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

Step 3.7

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

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Step 3.7.1

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

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Step 3.7.1.1

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

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

Step 3.7.1.2

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

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

Step 3.7.2

Add $1$ and $2$ .

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

Step 3.8

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

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

Step 4

Rewrite $\frac{\cos^{3} (x)}{\sin^{3} (x)}$ as $\cot^{3} (x)$ .

$\cot^{3} (x)$

Step 5

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

$\frac{\csc (x) \cot (x)}{\sec (x) \tan (x)} = \cot^{3} (x)$ is an identity