Trigonometry Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

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 $b = \tan (x)$ .

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

Step 2

Simplify the denominator.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Simplify.

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Step 2.4.1

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

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

Step 2.4.2

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 $b = \tan (x)$ .

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

Step 2.5

Rearrange terms.

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

Step 2.6

Apply pythagorean identity.

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

Step 3

Simplify terms.

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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.

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

Step 3.1.2

Rewrite the expression.

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

Step 3.2

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

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Step 3.2.1

Cancel the common factor.

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

Step 3.2.2

Rewrite the expression.

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

Step 3.3

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

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

Step 3.4

Rewrite $\frac{1^{2}}{\sec^{2} (x)}$ as ${(\frac{1}{\sec (x)})}^{2}$ .

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

Step 4

Rewrite $\sec (x)$ in terms of sines and cosines.

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

Step 5

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (x)}$ .

${(1 \cos (x))}^{2}$

Step 6

Multiply $\cos (x)$ by $1$ .

$\cos^{2} (x)$