SSC Sample Paper Mock Test-14 SSC CGL Tear-II Paper-1

If \[{{\sin }^{2}}x+{{\sin }^{2}}y+{{\sin }^{2}}z=\,\,{{(\sin x+\sin y+\sin z)}^{2}},\]then which of the following expressions must necessarily vanish?

A) \[-\,\,1\]

B) 2

C) 1

D) 0

Correct Answer: D

Solution :

\[{{\sin }^{2}}x+{{\sin }^{2}}y+{{\sin }^{2}}z={{(\sin x+\sin y+\sin z)}^{2}}\]

\[[\because {{(a+b+c)}^{2}}={{a}^{2}}+{{b}^{2}}+{{c}^{2}}+2\,\,(ab+bc+ca)]\]

\[\Rightarrow \]\[\sin x\sin y+\sin y\sin \,\,z+sin\,\,z\,\,sin\,\,x=0\]

Dividing both sides by \[x\sin y\sin z,\]

\[\Rightarrow \] \[\frac{1}{\sin x}+\frac{1}{\sin y}+\frac{1}{\sin \,\,z}=0\]