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

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

Start on the left side.

$\frac{1}{\sec (x) + \tan (x)}$

Convert to sines and cosines.

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

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

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

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

Simplify.

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Combine the numerators over the common denominator.

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

Multiply the numerator by the reciprocal of the denominator.

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

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

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

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

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

Combine.

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

Simplify numerator.

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Apply the distributive property.

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

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

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

Simplify each term.

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

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.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

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

Apply pythagorean identity.

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

Simplify.

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Factor $\cos (x)$ out of $\cos (x) - \cos (x) \sin (x)$ .

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Multiply by $1$ .

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

Factor $\cos (x)$ out of $- \cos (x) \sin (x)$ .

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

Factor $\cos (x)$ out of $\cos (x) \cdot 1 + \cos (x) (- \sin (x))$ .

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

Cancel the common factors.

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

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

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

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

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