Verify the Identity (cot(x)^2)/(csc(x))=csc(x)-sin(x)

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

Start on the left side.

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

Apply Pythagorean identity in reverse.

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

Convert to sines and cosines.

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

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

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

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

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

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

Simplify.

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Multiply the numerator by the reciprocal of the denominator.

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

One to any power is one.

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

Apply the distributive property.

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

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

$\csc (x) - \sin (x)$

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

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

Verify the Identity (cot(x)^2)/(csc(x))=csc(x)-sin(x)

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