Question

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

• A. $(1,0,0)$
• B. $(0,1,0)$
• C. $(\frac{1}{2},\frac{1}{2},\frac{2}{4})$
• D. $(\frac{1}{2},\frac{3}{4},\frac{1}{2})$

Answer 1

Answer: (A)

$(1,0,0)$

$\sin x=\cos y\Rightarrow {\sin }^{2}x={\cos }^{2}y=1-{\sin }^{2}y\Rightarrow u=1-v\sqrt{6}\sin y=\tan z\Rightarrow 6{\sin }^{2}y={\tan }^{2}z\Rightarrow 6v=\frac{w}{1-w}2\sin z=\sqrt{3}\cos x\Rightarrow 4{\sin }^{2}z=3{\cos }^{2}x\Rightarrow w=3\left(\frac{1-u}{4}\right)u=1\Rightarrow v=0,w=0$