Question

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

• A. $-2$
• B. $\sqrt{\frac{5}{2}}-1$
  
• C. $2$
• D. $5$

Answer 1

Answer: (B)

$\sqrt{\frac{5}{2}}-1$

${({\sin }^{2}x+{\cos }^{2}y)}^2-2{\sin }^{2}x{\cos }^{2}y+2=4\sin x\cos y.$

$1-2{\sin }^{2}x{\cos }^{2}y-4\sin x\cos y+2=0$

${(\sin x\cos y+1)}^2=\frac{5}{2}$

$\sin x\cos y=\sqrt{\frac{5}{2}}-1.$