If n is the number of solutions of the equation 2 x(4 ((π)/(4) + x)...

$2\cos x\left(4\sin \left(\dfrac{\pi}{4} + x\right)\sin \left(\dfrac{\pi}{4} - x\right) - 1\right) = 1$

$⇒ 2\cos x\left(2\cos \left(2x\right) - 2\cos \left(\dfrac{\pi}{2}\right) - 1\right) = 1$

$⇒ 2\cos x\left(2 . \left(2\cos^{2}x - 1\right) - 1\right) = 1$

$⇒ 2\cos x\left(4\cos^{2}x - 3\right) = 1$

$⇒ 4\cos^{3}x - 3\cos x = \dfrac{1}{2}$

$⇒ \cos 3x = \dfrac{1}{2}$

$⇒ 3x = \dfrac{\pi}{3}, \dfrac{5\pi}{3}, \dfrac{7\pi}{3}$

$⇒ x = \dfrac{\pi}{9}, \dfrac{5\pi}{9}, \dfrac{7\pi}{9}$

No. of solutions $= n = 3$

Sum of solutions $= S = \dfrac{\pi}{9} + \dfrac{5\pi}{9} + \dfrac{7\pi}{9} = \dfrac{13\pi}{9}$