Question

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

Answer 1

We have, $sin^{4}x=1+tan^{4}x$
$\therefore 0 \leq \sin ^{4} x \leq 1, (\because \sin x \in[-1,1])$
and $1 \leq 1+\tan ^{4} x<\infty, (\because \tan x \in R)$
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.