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

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Solution

The correct option is D 0
$s i n^{3} x . c o s x + s i n^{2} x . c o s^{2} x + s i n x . c o s^{3} x = 1$

$\Rightarrow s i n x . c o s x (s i n^{2} x + s i n x . c o s x + c o s^{2} x) = 1$

$\Rightarrow s i n x . c o s x (1 + s i n x . c o s x) = 1$

$\Rightarrow 4. s i n x . c o s x (1 + s i n x . c o s x) = 4$ .......... multiplied both side by 4

$\Rightarrow s i n 2 x (2 + s i n 2 x) = 4$
$\Rightarrow s i n^{2} 2 x + 2 s i n 2 x - 4 = 0$

Let $s i n 2 x = u$

$\Rightarrow u^{2} + 2 u - 4$

$∴ sin 2 x = - 1 \pm \sqrt{5}$

since both roots are beyond the range of $s i n x$ ,

therefore there is no solution.

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