Number of solutions of the equation $x^{2} + 4 x + s i n^{2} x + 4 c o s e c^{2} x = 0$ in $[0, 2 π]$ is /are___

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Solution

$x^{2} + 4 x + 4 + 4 c o t^{2} x + s i n^{2} x = 0 (x + 2)^{2} + 4 c o t^{2} x + s i n^{2} x = 0 (a)^{2} + (b)^{2} + (c)^{2} = 0 \Rightarrow a = b = c = 0$
Hence no solution.

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