Simplify (1+2tan(a)^2+tan(a)^4)/(sec(a)^4)

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

Simplify the numerator.

Factor using the perfect square rule.

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

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

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

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

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

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

Rewrite the polynomial.

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

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

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

Rearrange terms.

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

Apply pythagorean identity.

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

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

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

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

Multiply $2$ by $2$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

$1$