The number of solutions for the equation ^ 3 xcosx + ^ 2 x ^ 2 x +...

$\sin^{3}xcosx + \sin^{2}x\cos^{2}x + \sin x\cos^{3}x = 1$

$⇒ \sin x \cos x\left(\sin^{2}x + \sin x \cos x + \cos^{2}x\right) = 1$

$⇒ \dfrac{\sin 2x}{2}\left(1 + \dfrac{\sin 2x}{2}\right) = 1$

$⇒ \sin 2x\left(2 + \sin 2x\right) = 4$

$⇒ \sin^{2}2x + 2\sin 2x - 4 = 0$

$⇒ \sin 2x = \dfrac{- 2 \pm \sqrt{4 + 16}}{2}$

$⇒ \sin 2x = - 1 \pm \sqrt{5}$

This is not possible. \left(Since - 1 \le \sin 2x \le 1\right)$</p><p>Hence the given equation has no solution.</p>$