Verify the Identity cot(x)(cot(x)+tan(x))=csc(x)^2

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Trigonometry

$\cot (x) (\cot (x) + \tan (x)) = \csc^{2} (x)$

Start on the left side.

$\cot (x) (\cot (x) + \tan (x))$

Apply the distributive property.

$\cot (x) \cot (x) + \cot (x) \tan (x)$

Multiply $\cot (x) \cot (x)$ .

$\cot^{2} (x) + \cot (x) \tan (x)$

Apply Pythagorean identity in reverse.

$\csc^{2} (x) - 1 + \cot (x) \tan (x)$

Convert to sines and cosines.

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

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

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

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

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

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

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

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

Simplify.

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Simplify each term.

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One to any power is one.

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

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

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Cancel the common factor.

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

Rewrite the expression.

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

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

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Cancel the common factor.

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

Rewrite the expression.

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

Add $- 1$ and $1$ .

$\frac{1}{{\sin (x)}^{2}} + 0$

Add $\frac{1}{{\sin (x)}^{2}}$ and $0$ .

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

Now consider the right side of the equation.

$\csc^{2} (x)$

Convert to sines and cosines.

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

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

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

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

One to any power is one.

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

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

$\cot (x) (\cot (x) + \tan (x)) = \csc^{2} (x)$ is an identity

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