Question

The number of solutions of $\sin ^{7} x+\cos ^{7} x=1, x \in[0,4 \pi]$ is equal to

• A. 11
• B. 7
• C. 5
• D. 9

Answer 1

Answer: (C)

$\sin ^{7} x \leq \sin ^{2} x \leq \lambda 1$
and $\cos ^{7} x \leq \cos ^{2} x \leq 1$
also $\sin ^{2} x+\cos ^{2} x=1$
$\Rightarrow$ equality must hold for (1) $\&(2)$
$\Rightarrow \sin ^{7} x=\sin ^{2} x \& \cos ^{7}=\cos ^{2} x$
$\Rightarrow \sin x=0 \& \cos x=1$
or
$\cos x=0 \& \sin x=1$
$\Rightarrow x=0,2 \pi, 4 \pi, \frac{\pi}{2}, \frac{5 \pi}{2}$
$\Rightarrow 5$ solutions