Question

If $A+B+C=\pi$, then find the value of
$\frac{\cos A}{\sin B\sin C}+\frac{\cos B}{\sin A\sin C}+\frac{cosC}{sinBsinA}$ is

• A.
  0
• B.
  1
• C.
  2
• D.
  4

Answer 1

The correct option is C

2
Here, Given $A+B+C=\pi .$
Now, we can write the expression
$\frac{\cos A}{\sin B\sin C}+\frac{\cos B}{\sin A\sin C}+\frac{cosC}{sinBsinA}$ as
$\frac{\cos A\sin A+\cos B\sin B+\cos C\sin C}{\sin A\sin B\sin C}=\frac{1}{2}\left(\frac{2\cos A\sin A+2\cos B\sin B+2\cos C\sin C}{\sin A\sin B\sin C}\right)=\frac{1}{2}\left(\frac{\sin 2A+\sin 2B+\sin 2C}{\sin A\sin B\sin C}\right)$
Now, we know the expression
$\mathrm{when}A+B+C=\pi \Rightarrow \sin 2A+\sin 2B+\sin 2C=4\sin A\sin B\sin C$
Substituting the above equation, we get
$\frac{1}{2}\left(\frac{\sin 2A+\sin 2B+\sin 2C}{\sin A\sin B\sin C}\right)=\frac{4}{2}\left(\frac{\sin A\sin B\sin C}{\sin A\sin B\sin C}\right)=2$
Thus, Option c. is correct.