Question

If $\sin x=2\cos y,\sqrt{6}\sin y=\tan z$ and $2\sin z=\sqrt{3}\cos x,u,v,w$ denotes, respectively,${\sin }^{2}x,{\sin }^{2}y,{\sin }^{2}z,$ then the value of $u+v+w$ is

• A. $2$
  
• B. $3$
  
• C. $4$
  
• D. None of these
  

Answer 1

The correct option is D None of these

$u+v+w={\sin }^2\left(x\right)+{\sin }^{2}y+{\sin }^{2}z$

$={\sin }^2\left(x\right)+{\sin }^{2}y+\frac{3}{4}{\cos }^{2}x$

$=\left[{\sin }^2\right(x)+\frac{3}{4}{\cos }^{2}x]+{\sin }^{2}y$

$=[\frac{{\sin }^2\left(x\right)}{4}+\frac{3{\sin }^2\left(x\right)}{4}+\frac{3}{4}{\cos }^{2}x]+{\sin }^2\left(y\right)$

$=\frac{{\sin }^2\left(x\right)}{4}+\frac{3}{4}+1-{\cos }^{2}y$

$=\frac{{\sin }^2\left(x\right)}{4}+\frac{3}{4}+1-\frac{{\sin }^2\left(x\right)}{4}$

$=\frac{7}{4}$