Trigonometry Examples

$\frac{4 \cos (x) - 4 \cos^{3} (x)}{\sin (2 x)}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $4 \cos (x)$ out of $4 \cos (x) - 4 \cos^{3} (x)$ .

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Step 1.1.1

Factor $4 \cos (x)$ out of $4 \cos (x)$ .

$\frac{4 \cos (x) (1) - 4 \cos^{3} (x)}{\sin (2 x)}$

Step 1.1.2

Factor $4 \cos (x)$ out of $- 4 \cos^{3} (x)$ .

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

Step 1.1.3

Factor $4 \cos (x)$ out of $4 \cos (x) (1) + 4 \cos (x) (- \cos^{2} (x))$ .

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

Step 1.2

Rewrite $1$ as $1^{2}$ .

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

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 $a = 1$ and $b = \cos (x)$ .

$\frac{4 \cos (x) (1 + \cos (x)) (1 - \cos (x))}{\sin (2 x)}$

Step 2

Apply the sine double-angle identity.

$\frac{4 \cos (x) (1 + \cos (x)) (1 - \cos (x))}{2 \sin (x) \cos (x)}$

Step 3

Cancel the common factor of $4$ and $2$ .

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Step 3.1

Factor $2$ out of $4 \cos (x) (1 + \cos (x)) (1 - \cos (x))$ .

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

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $2$ out of $2 \sin (x) \cos (x)$ .

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 4

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

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Step 4.1

Cancel the common factor.

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

Step 4.2

Rewrite the expression.

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

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