Trigonometry Examples

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

Step 1

Simplify the left side.

Tap for more steps...

Step 1.1

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

Tap for more steps...

Step 1.1.1

Simplify the expression.

Tap for more steps...

Step 1.1.1.1

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

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

Step 1.1.1.2

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

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

Step 1.1.2

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

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

Step 1.1.3

Rearrange terms.

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

Step 1.1.4

Apply pythagorean identity.

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

Step 1.1.5

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

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

Step 1.1.6

Apply the cosine double-angle identity.

$\cos (2 x) = \cos (2 x)$

Step 2

Move all terms containing $\cos (2 x)$ to the left side of the equation.

Tap for more steps...

Step 2.1

Subtract $\cos (2 x)$ from both sides of the equation.

$\cos (2 x) - \cos (2 x) = 0$

Step 2.2

Subtract $\cos (2 x)$ from $\cos (2 x)$ .

$0 = 0$

Step 3

Since $0 = 0$ , the equation will always be true for any value of $x$ .

All real numbers

Step 4

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$