If n is the number of solutions of the equation 2 cos x 4 sin π 4 + x sin π 4 - x - 1 = 1 , x…

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

$\Rightarrow 2\cos x\left(2\cos \left(2x\right)-2\cos \left(\frac{\pi }{2}\right)-1\right)=1$

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

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

$\Rightarrow 4{\cos }^{3}x-3\cos x=\frac{1}{2}$

$\Rightarrow \cos 3x=\frac{1}{2}$

$\Rightarrow 3\mathrm{x}=\frac{\pi }{3},\frac{5\pi }{3},\frac{7\pi }{3}$

$\Rightarrow \mathrm{x}=\frac{\pi }{9},\frac{5\pi }{9},\frac{7\pi }{9}$

No. of solutions $=n=3$

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