Question

Assertion :In a $\Delta$ABC, $(a+b+c)(\tan \frac{A}{2}+\tan \frac{B}{2})=2c\cot \frac{C}{2}$ Reason: In a $\Delta$ABC $a=b\cos C+c\cos B$

• 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 B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

Assertion:
Consider, $(a+b+c)[\tan \frac{A}{2}+\tan \frac{B}{2}]$
$=(a+b+c)\tan (\frac{A}{2}+\frac{B}{2})[1-\tan \frac{A}{2}\tan \frac{B}{2}]$
$=2s\tan \left(\frac{\pi -C}{2}\right)[1-\sqrt{\frac{(s-b)(s-c)}{s(s-a)}}\sqrt{\frac{(s-a)(s-c)}{s(s-b)}}]$
$=\cot \frac{C}{2}[2s-2s\frac{(s-c)}{s}]$
$=2c\cot \frac{C}{2}$
Reason:
For any triangle ABC,
$a=b\cos C+c\cos B$
$b=a\cos C+c\cos A$
$c=a\cos B+b\cos A$
Hence, reason is correct but not a correct explanation of assertion.