Question

In $\Delta$ $ABC$ the sides opposite to angles $A,B,C$ are denoted by $a,b,c$ respectively.

If$\frac{a}{\cos A}=\frac{b}{\cos B},$ then which of the follwoing is true?

• A. $2\sin A\sin B\sin C=1$
• B. ${\sin }^{2}A+{\sin }^{2}B+={\sin }^{2}C$
• C. $2\sin A\cos B=\sin C$
• D. none of these

Answer 1

The correct option is C $2\sin A\cos B=\sin C$

Using sine rule

$\frac{a}{\cos A}=\frac{b}{\cos B}\frac{k\sin A}{\cos A}=\frac{k\sin B}{\cos B}\Rightarrow A=BA+B+C=\pi C=\pi -A-B\sin C=\sin (\pi -A-B)\sin C=\sin (A+B)\sin C=\sin A\cos B+\cos A\sin B$

Now $A=B$

$\Rightarrow \sin C=2\sin A\cos B$