The number of solutions of ^ 7 x + ^ 7 x = 1, x ∈ [ 0, 4π ] is...

We know that $0 \le \sin^{2}x \le 1$ and $0 \le \cos^{2}x \le 1$

Also, we know for a number which lie between $0$ to $1,$ higher the power of the number the less is its value.

$⇒ \sin^{7}x \le \sin^{2}x \le 1 \ldots \left(1\right)$

and $\cos^{7}x \le \cos^{2}x \le 1 \ldots \left(2\right)$

Also, we know that $\sin^{2}x + \cos^{2}x = 1$

This means the equality must hold for \left(1\right)$and \left(2\right)$

$⇒ \sin^{7}x = \sin^{2}x$ and $\cos^{7}x = \cos^{2}x$

$⇒ \sin^{2}x\left(\sin^{5}x - 1\right) = 0$ and $\cos^{2}x\left(\cos^{5}x - 1\right) = 0$

$⇒ \sin x = 0$ or $\sin x = 1$ and $\cos x = 0$ or $\cos x = 1$$⇒ x = 0, 2\pi, 4\pi, \dfrac{\pi}{2}, \dfrac{5\pi}{2} .$

Thus, there are total $5$ solutions.