Trigonometry Examples

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

Start on the right side.

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

Apply Pythagorean identity in reverse.

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

Convert to sines and cosines.

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

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

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

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

Simplify.

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Simplify each term.

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One to any power is one.

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

Apply the distributive property.

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

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

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

Multiply $2$ by $- 1$ .

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

Subtract $2$ from $1$ .

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

Reorder terms.

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

Rewrite $- 1 + \frac{2}{\cos^{2} (x)}$ as $2 \sec^{2} (x) - 1$ .

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

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

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