Let S be the sum of all solutions (in radians) of the equation ^ 4...

Given that: $\sin^{4} \theta + \cos^{4} \theta - sin \theta cos \theta = 0; \theta \in [ 0, 4\pi ]$

$⇒ 1 - 2\sin^{2} \theta\cos^{2} \theta - sin \theta cos \theta = 0$

$⇒ 2 - \sin^{2} 2\theta - \sin 2\theta = 0$

$⇒ \sin^{2} 2\theta + \sin 2\theta - 2 = 0$

$⇒ \sin 2\theta = 1; 2\theta \in [ 0, 8\pi ]$

$\theta = \dfrac{\pi}{4}, \dfrac{5\pi}{4}, \dfrac{9\pi}{4}, \dfrac{13\pi}{4}$

$⇒$Sum of the solutions $= S = \dfrac{28\pi}{4}$
Then value of $\dfrac{8S}{\pi} = 8 \times \left(\dfrac{28\pi}{4\pi}\right) = 56$