Trigonometry Examples

$5 \cos^{2} (B) - 5 \sin^{2} (B) = 10 \cos^{2} (B) - 5$

Step 1

Start on the left side.

$5 \cos^{2} (B) - 5 \sin^{2} (B)$

Step 2

Apply Pythagorean identity in reverse.

$5 (1 - \sin^{2} (B)) - 5 \sin^{2} (B)$

Step 3

Simplify.

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Step 3.1

Simplify each term.

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Step 3.1.1

Apply the distributive property.

$5 \cdot 1 + 5 (- {\sin (B)}^{2}) - 5 {\sin (B)}^{2}$

Step 3.1.2

Multiply $5$ by $1$ .

$5 + 5 (- {\sin (B)}^{2}) - 5 {\sin (B)}^{2}$

Step 3.1.3

Multiply $- 1$ by $5$ .

$5 - 5 {\sin (B)}^{2} - 5 {\sin (B)}^{2}$

Step 3.2

Subtract $5 {\sin (B)}^{2}$ from $- 5 {\sin (B)}^{2}$ .

$5 - 10 \sin^{2} (B)$

Step 4

Apply Pythagorean identity in reverse.

$5 - 10 (1 - \cos^{2} (B))$

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.

$5 - 10 \cdot 1 - 10 (- {\cos (B)}^{2})$

Step 5.1.2

Multiply $- 10$ by $1$ .

$5 - 10 - 10 (- {\cos (B)}^{2})$

Step 5.1.3

Multiply $- 1$ by $- 10$ .

$5 - 10 + 10 {\cos (B)}^{2}$

Step 5.2

Subtract $10$ from $5$ .

$- 5 + 10 \cos^{2} (B)$

Step 6

Rewrite $- 5 + 10 \cos^{2} (B)$ as $10 \cos^{2} (B) - 5$ .

$10 \cos^{2} (B) - 5$

Step 7

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

$5 \cos^{2} (B) - 5 \sin^{2} (B) = 10 \cos^{2} (B) - 5$ is an identity