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

$\sec (x) - \tan (x) \sin (x) = \cos (x)$

Start on the left side.

$\sec (x) - \tan (x) \sin (x)$

Convert to sines and cosines.

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Apply pythagorean identity.

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

Cancel the common factor of ${\cos (x)}^{2}$ and $\cos (x)$ .

$\cos (x)$

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

$\sec (x) - \tan (x) \sin (x) = \cos (x)$ is an identity

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

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