Simplify (cos(x)^4-sin(x)^4)/(cos(x)^2)

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

Step 1

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

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

Step 2

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

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

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

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

Step 4

Simplify.

Rearrange terms.

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

Apply pythagorean identity.

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

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

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

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 $b = \sin (x)$ .

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

Simplify (cos(x)^4-sin(x)^4)/(cos(x)^2)

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