Question

How do you simplify $se{c}^{4}x-{\tan }^{4}x$?

Answer 1

Simplify the expression:

Given expression is $se{c}^{4}x-{\tan }^{4}x...\left(1\right)$.

Note that,

$\begin{array}{rcl}secx& =& \frac{1}{\cos x}\\ \tan x& =& \frac{\sin x}{\cos x}\\ {\cos }^{2}x& =& 1-{\sin }^{2}x\end{array}$

Apply $a^2-b^2=\left(a+b\right)(a-b)$ in equation $\left(1\right)$

$\begin{array}{rcl}se{c}^{4}x-{\tan }^{4}x& =& \left(se{c}^{2}x-{\tan }^{2}x\right)\left(se{c}^{2}x+{\tan }^{2}x\right)\\ & =& \left(\frac{1}{{\cos }^{2}x}-\frac{{\sin }^{2}x}{{\cos }^{2}x}\right)\left(\left(1+{\tan }^{2}x\right)+{\tan }^{2}x\right)\\ & =& \left(\frac{1-{\sin }^{2}x}{{\cos }^{2}x}\right)\left(1+2{\tan }^{2}x\right)\\ & =& \left(\frac{{\cos }^{2}x}{{\cos }^{2}x}\right)\left(1+2{\tan }^{2}x\right)\\ & =& 1\left(1+2{\tan }^{2}x\right)\\ & =& 1+2{\tan }^{2}x\end{array}$

Hence, $se{c}^{4}x-{\tan }^{4}x=1+2{\tan }^{2}x$.