Verify the Identity cos(x)^2(1+tan(x)^2)=1

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

Start on the left side.

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

Apply Pythagorean identity.

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Rearrange terms.

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

Apply pythagorean identity.

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

Convert to sines and cosines.

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

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

Apply the product rule to $\frac{1}{\cos (x)}$ .

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

Simplify.

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One to any power is one.

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

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

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Cancel the common factor.

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

Rewrite the expression.

$1$

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

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

Verify the Identity cos(x)^2(1+tan(x)^2)=1

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