The number of integral values of n such that sinx(sinx+cosx)=n has atleast one real solution is

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Solution

The correct option is D 2
$s i n x (s i n x + c o s x) = n \Rightarrow s i n^{2} x + s i n x c o s x = n \Rightarrow \frac{1 - c o s 2 x}{2} + \frac{s i n 2 x}{2} = n \Rightarrow s i n 2 x - c o s 2 x = 2 n - 1 \Rightarrow - \sqrt{2} \leq 2 n - 1 \leq \sqrt{2} \Rightarrow \frac{1 - \sqrt{2}}{2} \leq n \leq \frac{1 + \sqrt{2}}{2} ∴ n = 0, 1$

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