Trigonometry Examples

$\cos^{2} (b) - \sin^{2} (b) = 2 \cos^{2} (b) - 1$

Step 1

Start on the left side.

$\cos^{2} (b) - \sin^{2} (b)$

Step 2

Apply Pythagorean identity in reverse.

$1 - \sin^{2} (b) - \sin^{2} (b)$

Step 3

Subtract ${\sin (b)}^{2}$ from $- {\sin (b)}^{2}$ .

$1 - 2 \sin^{2} (b)$

Step 4

Apply Pythagorean identity in reverse.

$1 - 2 (1 - \cos^{2} (b))$

Step 5

Simplify.

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Step 5.1

Simplify each term.

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Step 5.1.1

Apply the distributive property.

$1 - 2 \cdot 1 - 2 (- {\cos (b)}^{2})$

Step 5.1.2

Multiply $- 2$ by $1$ .

$1 - 2 - 2 (- {\cos (b)}^{2})$

Step 5.1.3

Multiply $- 1$ by $- 2$ .

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

Step 5.2

Subtract $2$ from $1$ .

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

Step 6

Rewrite $- 1 + 2 \cos^{2} (b)$ as $2 \cos^{2} (b) - 1$ .

$2 \cos^{2} (b) - 1$

Step 7

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

$\cos^{2} (b) - \sin^{2} (b) = 2 \cos^{2} (b) - 1$ is an identity