Question

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

• A. $5$
• B. $7$
• C. $11$
• D. $9$

Answer 1

The correct option is A $5$

${\sin }^{7}x+{\cos }^{7}x=1$
As ${\sin }^{7}x+{\cos }^{7}x\le {\sin }^{2}x+{\cos }^{2}x\le 1$
The equation gives solution only when one of $\sin x,\cos x$ is unity and other vanishes.
i.e., $x=0,\frac{\pi }{2},2\pi ,\frac{5\pi }{2},4\pi$