Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Factor.

Tap for more steps...

Step 2.1

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

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

Step 2.2

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

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

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 $a = \cos^{2} (x)$ and $b = \sin^{2} (x)$ .

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

Step 2.4

Simplify.

Tap for more steps...

Step 2.4.1

Rearrange terms.

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

Step 2.4.2

Apply pythagorean identity.

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

Step 2.4.3

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

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

Step 2.4.4

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

$(\cos (x) + \sin (x)) (\cos (x) - \sin (x))$

Step 3

Apply the distributive property.

$(\cos (x) + \sin (x)) \cos (x) + (\cos (x) + \sin (x)) (- \sin (x))$

Step 4

Simplify.

Tap for more steps...

Step 4.1

Simplify each term.

Tap for more steps...

Step 4.1.1

Apply the distributive property.

$\cos (x) \cos (x) + \sin (x) \cos (x) + (\cos (x) + \sin (x)) (- \sin (x))$

Step 4.1.2

Multiply $\cos (x) \cos (x)$ .

Tap for more steps...

Step 4.1.2.1

Raise $\cos (x)$ to the power of $1$ .

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

Step 4.1.2.2

Raise $\cos (x)$ to the power of $1$ .

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

Step 4.1.2.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 4.1.2.4

Add $1$ and $1$ .

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

Step 4.1.3

Apply the distributive property.

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

Step 4.1.4

Multiply $\sin (x) (- \sin (x))$ .

Tap for more steps...

Step 4.1.4.1

Raise $\sin (x)$ to the power of $1$ .

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

Step 4.1.4.2

Raise $\sin (x)$ to the power of $1$ .

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

Step 4.1.4.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 4.1.4.4

Add $1$ and $1$ .

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

Step 4.2

Add $\cos (x) \sin (x)$ and $\cos (x) (- \sin (x))$ .

Tap for more steps...

Step 4.2.1

Reorder $\cos (x)$ and $- 1$ .

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

Step 4.2.2

Subtract $\cos (x) \sin (x)$ from $\cos (x) \sin (x)$ .

${\cos (x)}^{2} + 0 - {\sin (x)}^{2}$

Step 4.3

Add ${\cos (x)}^{2}$ and $0$ .

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

Step 5

Apply Pythagorean identity in reverse.

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

Step 6

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

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

Step 7

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

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