Trigonometry Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

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

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

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

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

Step 1.4

Simplify.

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Step 1.4.1

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

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

Step 1.4.2

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 $b = \tan (x)$ .

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

Step 1.5

Rearrange terms.

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

Step 1.6

Apply pythagorean identity.

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

Step 2

Simplify the denominator.

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Step 2.1

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

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

Step 2.2

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 $b = \tan (x)$ .

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

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

Cancel the common factor of $1 + \tan (x)$ .

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Step 3.1.1

Cancel the common factor.

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

Step 3.1.2

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $1 - \tan (x)$ .

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Step 3.2.1

Cancel the common factor.

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

Step 3.2.2

Divide $\sec^{2} (x)$ by $1$ .

$\sec^{2} (x)$