Question

Find the total number of integral values of $n$ so that $\sin x(\sin x+\cos x)=n$ has at least one solution

Answer 1

$\sin x(\sin x+\cos x)=n$

${\sin }^{2}x+\sin x\cos x=n$

$\frac{1-\cos 2x}{2}+\frac{\sin 2x}{2}=n$

$\sin 2x-\cos 2x=2n-1$

$\Rightarrow$ $-\sqrt{2}\le 2n-1\le \sqrt{2}$

$\frac{1-\sqrt{2}}{2}\le n\le \frac{1+\sqrt{2}}{2}$

$n=0,1$
Ans: 2