Trigonometry Examples

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

Step 1

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

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Step 1.1

Simplify the expression.

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Step 1.1.1

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

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

Step 1.1.2

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

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

Step 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 1.3

Rearrange terms.

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

Step 1.4

Apply pythagorean identity.

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

Step 1.5

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

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

Step 1.6

Apply the cosine double-angle identity.

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

Step 2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx 0, - 0.1, 0.1, \frac{1}{5}, - 0.3, 0.3, - \frac{2}{5}$

Step 3