Question

Assertion :$\sin {52}^o+\sin {78}^o+\sin {50}^o=4\cos {26}^o\cos {39}^o\cos {25}^o$. Reason: If $A+B+C=\pi$, then $\sin A+\sin B+\sin C=4\cos \left(\frac{A}{2}\right)\cos \left(\frac{B}{2}\right)\cos \left(\frac{C}{2}\right)$.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Reason:
$A+B+C=\pi \Rightarrow A+B=\pi -C\sin A+\sin B+\sin C=\sin A+\sin B+\sin (\pi -(A+B))=\sin A+\sin B+\sin (A+B)=2\sin \left(\frac{A+B}{2}\right)\cos \left(\frac{A-B}{2}\right)+2\sin \left(\frac{A+B}{2}\right)\cos \left(\frac{A+B}{2}\right)=2\sin \left(\frac{A+B}{2}\right)(\cos \left(\frac{A-B}{2}\right)+\cos \left(\frac{A+B}{2}\right))=2\sin (\frac{\pi }{2}-\frac{C}{2})\left(2\cos \frac{A}{2}\cos \frac{B}{2}\right)=4\cos \frac{A}{2}\cos \frac{B}{2}\cos \frac{C}{2}$

Assertion:
As $58+72+50=180$
U\sin g $\sin A+\sin B+\sin C=4\cos \frac{A}{2}\cos \frac{B}{2}\cos \frac{C}{2}$
We get,
$\sin 58+\sin 72+\sin 50=4\cos 29\cos 36\cos 25$