Mathematics - Trigonometric Ratios & Identities Question with Solution | TestHub

Because $\sin y\cos z=\{\sin (y+z)+\sin (y-z)\}/2,P\ge \frac{1}{2}\cos x\sin (\pi /2-x)=\frac{1}{2}{\cos }^{2}x\ge 1/8$; equality occurs at $(x,y,z)=(\pi /3,\pi /12,\pi /12)$. Also $\sin y\cos x\le \frac{1}{2}\sin (x+y)=\frac{1}{2}\cos z$, so $P\le \frac{1}{2}{\cos }^{2}z\le (2+\sqrt{3})/8$, attained at $(x,y,z)=(5\pi /24,5\pi /24,\pi /12)$.