Trigonometry Examples

$\cos^{2} (x) + \tan^{2} (x) \cos^{2} (x) = 1$

Start on the left side.

$\cos^{2} (x) + \tan^{2} (x) \cos^{2} (x)$

Apply Pythagorean identity in reverse.

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

Convert to sines and cosines.

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

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

Apply the product rule to $\frac{\sin (x)}{\cos (x)}$ .

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

Simplify.

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

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Cancel the common factor.

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

Rewrite the expression.

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

Add $- {\sin (x)}^{2}$ and ${\sin (x)}^{2}$ .

$1 + 0$

Add $1$ and $0$ .

$1$

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

$\cos^{2} (x) + \tan^{2} (x) \cos^{2} (x) = 1$ is an identity