Question

Find the integrals of the function :
$\tan 4x$

Answer 1

Simplifying $\tan 4x$.

$\tan 4x=\tan 2\left(2x\right)$

$=\frac{2\tan 2x}{1-{\tan }^{2}2x}$

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

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

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

$=\frac{4\tan x\left(1-{\tan }^{2}x\right)}{1+{\tan }^{4}x-6{\tan }^{2}x}$