Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Convert to sines and cosines.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

Apply Pythagorean identity in reverse.

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

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Rewrite $1$ as $1^{2}$ .

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

Step 4.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = \cos (x)$ .

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

Step 4.2

Write $1$ as a fraction with a common denominator.

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

Step 4.3

Combine the numerators over the common denominator.

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

Step 4.4

Simplify the numerator.

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

Step 5

Apply the cosine double-angle identity.

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

Step 6

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

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