Verify the Identity (csc(x))/(cot(x)+tan(x))=cos(x)

$\frac{\csc (x)}{\cot (x) + \tan (x)} = \cos (x)$

Start on the left side.

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

Convert to sines and cosines.

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

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

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

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

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

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

Simplify.

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Multiply the numerator by the reciprocal of the denominator.

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

Simplify the denominator.

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To write $\frac{\cos (x)}{\sin (x)}$ as a fraction with a common denominator, multiply by $\frac{\cos (x)}{\cos (x)}$ .

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

To write $\frac{\sin (x)}{\cos (x)}$ as a fraction with a common denominator, multiply by $\frac{\sin (x)}{\sin (x)}$ .

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

Write each expression with a common denominator of $\sin (x) \cos (x)$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

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

Combine.

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

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Multiply $\cos (x) \cos (x)$ .

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

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

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

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

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

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

Add $1$ and $1$ .

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

Multiply $\sin (x) \sin (x)$ .

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

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

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

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

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

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

Add $1$ and $1$ .

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

Multiply the numerator by the reciprocal of the denominator.

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

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

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

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

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

Apply pythagorean identity.

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

Divide $\cos (x)$ by $1$ .

$\cos (x)$

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

$\frac{\csc (x)}{\cot (x) + \tan (x)} = \cos (x)$ is an identity

Verify the Identity (csc(x))/(cot(x)+tan(x))=cos(x)

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