Verify the Identity (cot(x)^2)/(csc(x))=(1-sin(x)^2)/(sin(x))

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Trigonometry

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

Start on the right side.

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

Apply pythagorean identity.

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

Now consider the left side of the equation.

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

Convert to sines and cosines.

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

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

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

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

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

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

Simplify.

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Multiply the numerator by the reciprocal of the denominator.

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

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

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

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

$\frac{\cot^{2} (x)}{\csc (x)} = \frac{1 - \sin^{2} (x)}{\sin (x)}$ is an identity

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