Trigonometry Examples

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

Step 1

Simplify the numerator.

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Step 1.1

Use the double-angle identity to transform $\cos (2 x)$ to $2 \cos^{2} (x) - 1$ .

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Multiply $2$ by $- 1$ .

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

Step 1.4

Multiply $- 1$ by $- 1$ .

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

Step 1.5

Add $1$ and $1$ .

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

Step 1.6

Rewrite $2 - 2 \cos^{2} (x)$ in a factored form.

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Step 1.6.1

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

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Step 1.6.1.1

Factor $2$ out of $2$ .

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

Step 1.6.1.2

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

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

Step 1.6.1.3

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

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

Step 1.6.2

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

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

Step 1.6.3

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 $b = \cos (x)$ .

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

Step 2

Simplify the denominator.

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Step 2.1

Use the double-angle identity to transform $\cos (2 x)$ to $2 \cos^{2} (x) - 1$ .

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

Step 2.2

Subtract $1$ from $1$ .

$\frac{2 (1 + \cos (x)) (1 - \cos (x))}{0 + 2 \cos^{2} (x)}$

Step 2.3

Add $0$ and $2 \cos^{2} (x)$ .

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

Step 3

Cancel the common factor of $2$ .

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Step 3.1

Cancel the common factor.

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

Step 3.2

Rewrite the expression.

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