Verify the Identity sin(x)(cot(x)+tan(x))=sec(x)

$\sin (x) (\cot (x) + \tan (x)) = \sec (x)$

Start on the left side.

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

Convert to sines and cosines.

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

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

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

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

Apply the distributive property.

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

Simplify each term.

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

Apply Pythagorean identity in reverse.

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

Simplify.

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Simplify the numerator.

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Rewrite $1$ as $1^{2}$ .

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

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = \cos (x)$ .

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

To write $\frac{\cos (x)}{1}$ as a fraction with a common denominator, multiply by $\frac{\cos (x)}{\cos (x)}$ .

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

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

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Combine.

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

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

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

$\sec (x)$

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

$\sin (x) (\cot (x) + \tan (x)) = \sec (x)$ is an identity

Verify the Identity sin(x)(cot(x)+tan(x))=sec(x)

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