Algebra Examples

$\frac{1 + 2 \tan^{2} (a) + \tan^{4} (a)}{\sec^{4} (a)}$

Step 1

Simplify the numerator.

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Step 1.1

Factor using the perfect square rule.

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Step 1.1.1

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

$\frac{1^{2} + 2 \tan^{2} (a) + \tan^{4} (a)}{\sec^{4} (a)}$

Step 1.1.2

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

$\frac{1^{2} + 2 \tan^{2} (a) + {(\tan^{2} (a))}^{2}}{\sec^{4} (a)}$

Step 1.1.3

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 \tan^{2} (a) = 2 \cdot 1 \cdot \tan^{2} (a)$

Step 1.1.4

Rewrite the polynomial.

$\frac{1^{2} + 2 \cdot 1 \cdot \tan^{2} (a) + {(\tan^{2} (a))}^{2}}{\sec^{4} (a)}$

Step 1.1.5

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = 1$ and $b = \tan^{2} (a)$ .

$\frac{{(1 + \tan^{2} (a))}^{2}}{\sec^{4} (a)}$

Step 1.2

Rearrange terms.

$\frac{{(\tan^{2} (a) + 1)}^{2}}{\sec^{4} (a)}$

Step 1.3

Apply pythagorean identity.

$\frac{{(\sec^{2} (a))}^{2}}{\sec^{4} (a)}$

Step 1.4

Multiply the exponents in ${(\sec^{2} (a))}^{2}$ .

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Step 1.4.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{{\sec (a)}^{2 \cdot 2}}{\sec^{4} (a)}$

Step 1.4.2

Multiply $2$ by $2$ .

$\frac{\sec^{4} (a)}{\sec^{4} (a)}$

Step 2

Cancel the common factor of $\sec^{4} (a)$ .

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Step 2.1

Cancel the common factor.

$\frac{\sec^{4} (a)}{\sec^{4} (a)}$

Step 2.2

Rewrite the expression.

$1$