Trigonometry Examples

$\frac{\csc^{2} (x)}{\cot (x)} = \csc (x) \sec (x)$

Step 1

Start on the left side.

$\frac{\csc^{2} (x)}{\cot (x)}$

Step 2

Convert to sines and cosines.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

Simplify.

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.2

Combine.

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

Cancel the common factors.

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Step 3.3.2.1

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

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

Step 3.3.2.2

Cancel the common factor.

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

Step 3.3.2.3

Rewrite the expression.

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

Step 3.4

One to any power is one.

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

Step 4

Reorder terms.

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

Step 5

Rewrite $\frac{1}{\cos (x) \sin (x)}$ as $\csc (x) \sec (x)$ .

$\csc (x) \sec (x)$

Step 6

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

$\frac{\csc^{2} (x)}{\cot (x)} = \csc (x) \sec (x)$ is an identity