Question

The number of solution of the equation $1+\sin x{\sin }^2\frac{x}{2}=0$ in $[\pi ,\pi ]$ is-

• A. Zero
• B. $1$
• C. $2$
• D. $3$

Answer 1

The correct option is A Zero

Given

$1+\sin x{\sin }^2\frac{x}{2}=0$

$⟹\sin x{\sin }^2\frac{x}{2}=-1$

The range of $\sin$ is $[-1,1]$

So, to get $-1$ as product one of the multipliers must be $-1,1$

${\sin }^2\frac{x}{2}$ cannot be negative

$⟹\sin x=-1$

$⟹x=\frac{3\pi }{2}$

But ${\sin }^2\frac{3\pi }{4}\ne 1$

So there is no such solution.