Question

The number of real solutions of the equation ${\cos }^{5}x+{\sin }^{3}x=1$ in the interval $[0,2\pi ]$ is:

• A. $2$
• B. $1$
• C. $3$
• D. infinite

Answer 1

Answer: (B)

$1$

${\cos }^{5}x+{\sin }^{3}x=1$ , $x\in [0,2\pi ]$

At $x=0,{\sin }^{3}x=0,{\cos }^{5}x=1$

At $x=\frac{\pi }{2},{\sin }^{3}x=1,{\cos }^{5}x=0$

$\therefore {\sin }^{3}x+{\cos }^{5}x=1$ has same solutions.

of $\sin x+\cos x=1$

As powers of $\sin x$ and $\cos x$ are odd,

$\Rightarrow \sin x\ne -1,\cos x\ne -1$

$\therefore x=0,\frac{\pi }{2},2\pi$ are possible solutions.