Question

Assertion (A): $\ln \Delta ABC,\sum \frac{\cos A}{\sin B\sin C}=2$.
Reason(R): $\ln \Delta ABC,\sin A+\sin B+\sin C=4\cos \frac{A}{2}\cos \frac{B}{2}\cos \frac{C}{2}$.

• A. A is true, R is true and R is correct explanation of A
• B. A is true, R is true and R is not correct explanation of A
• C. A is true, R is false
• D. A is false, R is true.

Answer 1

The correct option is B A is true, R is true and R is not correct explanation of A

$\frac{\cos A}{\sin B\sin C}+\frac{\cos B}{\sin A\sin C}+\frac{\cos C}{\sin A\sin B}=2$

$\frac{\sin A\cos A+\sin B\cos B+\sin C\cos C}{\sin A\sin B\sin C}=2$

$\sin 2A+\sin 2B+\sin 2C=4\sin A\sin B\sin C$

$2\sin (A+B)\cos (A-B)+\sin 2C=4\sin A\sin B\sin C$

$2\sin (A+B)\cos (A-B)+\sin 2C=4\sin A\sin B\sin C$

$\cos (A-B)+\cos C=2\sin A\sin B$

$\cos (A-B)-\cos (A+B)$

$=2\sin A\sin B$

Hence, proved.

$2\sin \left(\frac{A+B}{2}\right)\cos \left(\frac{A-B}{2}\right)+2\sin \frac{C}{2}\cos \frac{C}{2}$

$2\cos \frac{C}{2}[\cos \left(\frac{A-B}{2}\right)+\cos \left(\frac{A+B}{2}\right)]$

$=4\cos \frac{A}{2}\cos \frac{B}{2}\cos \frac{C}{2}$ hence proved.

Therefore, A is true, R is true and R is not correc explanation of A.