Question

Find the number of solutions of the equation ${\sin }^{4}x=1+{\tan }^{4}x$ in $\left(0,4\pi \right)$ .

Answer 1

Answer: 0

We have, $si{n}^{4}x=1+ta{n}^{4}x$
$\therefore 0\le {\sin }^{4}x\le 1,(\because \sin x\in [-1,1])$
and $1\le 1+{\tan }^{4}x<\infty ,(\because \tan x\in R)$
So, $LHS=RHS=1$
$\Rightarrow {\sin }^{4}x=1$ and $1+{\tan }^{4}x=1$
$\Rightarrow {\sin }^{2}x=1$ and $\tan x=0$
So, This is not possible, so the given equation have no solution.