Question

lf ${\sin }^{4}x+{\cos }^{4}y+2=$
4 $\sin x\cos y,0<x,y\le \frac{\pi }{2}$ , then $\sin x+\cos y$ equal to

• A. $0$
• B. $-2$
• C. $1$
• D. $2$

Answer 1

Answer: (D)

$2$

apply AM-GM for ${\sin }^{4}x,{\cos }^{4}y,1,1$
we get $\frac{{\sin }^{4}x+{\cos }^{4}y+1+1}{4}>({\sin }^{4}x{\cos }^{4}y{)}^{\frac{1}{4}}$
this gives ${\sin }^{4}x+{\cos }^{4}y+2>4\sin x\cos y$
here as given , equality occurs
so $\sin x=\cos y=1$
so $\sin x+\cos y=2$