Verify the Identity 1/(1-sin(x))-1/(1+sin(x))=2tan(x)sec(x)

$\frac{1}{1 - \sin (x)} - \frac{1}{1 + \sin (x)} = 2 \tan (x) \sec (x)$

Start on the left side.

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

Subtract fractions.

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

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

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

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

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

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Combine.

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

Combine.

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

Reorder the factors of $(1 - \sin (x)) (1 + \sin (x))$ .

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

Combine the numerators over the common denominator.

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

Simplify numerator.

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Simplify each term.

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Multiply $1 + \sin (x)$ by $1$ .

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

Apply the distributive property.

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

Multiply $- 1$ by $1$ .

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

Multiply $- - \sin (x)$ .

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Multiply $- 1$ by $- 1$ .

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

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

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

Subtract $1$ from $1$ .

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

Add $0$ and $\sin (x)$ .

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

Add $\sin (x)$ and $\sin (x)$ .

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

Simplify denominator.

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Expand $(1 + \sin (x)) (1 - \sin (x))$ using the FOIL Method.

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Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

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

Apply pythagorean identity.

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

Now consider the right side of the equation.

$2 \tan (x) \sec (x)$

Convert to sines and cosines.

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

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

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

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

Simplify.

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Combine $2$ and $\frac{\sin (x)}{\cos (x)}$ .

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

Multiply $\frac{2 \sin (x)}{\cos (x)} \cdot \frac{1}{\cos (x)}$ .

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

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

$\frac{1}{1 - \sin (x)} - \frac{1}{1 + \sin (x)} = 2 \tan (x) \sec (x)$ is an identity

Verify the Identity 1/(1-sin(x))-1/(1+sin(x))=2tan(x)sec(x)

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