Trigonometry Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

Simplify.

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Step 1.4.1

Apply pythagorean identity.

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

Step 1.4.2

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

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

Step 1.4.3

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

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

Step 2

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

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

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

Cancel the common factor of $\sin (x) + \cos (x)$ .

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Step 3.1.1

Cancel the common factor.

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

Step 3.1.2

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $\sin (x) - \cos (x)$ .

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Step 3.2.1

Cancel the common factor.

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

Step 3.2.2

Rewrite the expression.

$1$