Trigonometry Examples

$\sin^{4} (x) - \cos^{4} (x) = 2 \sin^{2} (x) - 1$

Step 1

Start on the left side.

$\sin^{4} (x) - \cos^{4} (x)$

Step 2

Simplify the expression.

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Step 2.1

Rewrite $\sin^{4} (x)$ as ${(\sin^{2} (x))}^{2}$ .

${(\sin^{2} (x))}^{2} - \cos^{4} (x)$

Step 2.2

Rewrite $\cos^{4} (x)$ as ${(\cos^{2} (x))}^{2}$ .

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

Step 2.3

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

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

Step 2.4

Apply pythagorean identity.

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

Step 2.5

Multiply $\sin^{2} (x) - \cos^{2} (x)$ by $1$ .

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

Step 3

Apply Pythagorean identity in reverse.

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

Step 4

Subtract ${\cos (x)}^{2}$ from $- {\cos (x)}^{2}$ .

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

Step 5

Apply Pythagorean identity in reverse.

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

Step 6

Simplify.

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Step 6.1

Simplify each term.

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Step 6.1.1

Apply the distributive property.

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

Step 6.1.2

Multiply $- 2$ by $1$ .

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

Step 6.1.3

Multiply $- 1$ by $- 2$ .

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

Step 6.2

Subtract $2$ from $1$ .

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

Step 7

Rewrite $- 1 + 2 \sin^{2} (x)$ as $2 \sin^{2} (x) - 1$ .

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

Step 8

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

$\sin^{4} (x) - \cos^{4} (x) = 2 \sin^{2} (x) - 1$ is an identity