Algebra Examples

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

Step 1

Rearrange terms.

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

Step 2

Apply pythagorean identity.

$\frac{1 - \tan^{2} (\frac{x}{2})}{\sec^{2} (\frac{x}{2})}$

Step 3

Simplify the numerator.

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Step 3.1

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

$\frac{1^{2} - \tan^{2} (\frac{x}{2})}{\sec^{2} (\frac{x}{2})}$

Step 3.2

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 $b = \tan (\frac{x}{2})$ .

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