Question

Assertion :In a triangle ABC if a, b, c are in A.P., then $\cot \frac{A}{2}\cot \frac{C}{2}=2$ Reason: Three numbers a, b, c are in A.P. if $a+c=2b$.

• 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 false but Reason is true

Answer 1

The correct option is D Assertion false but Reason is true

Using $tan\frac{A}{2}=\sqrt{\frac{(s-b)(s-c)}{s(s-a)}},tan\frac{B}{2}=\sqrt{\frac{(s-c)(s-a)}{s(s-b)}},tan\frac{C}{2}=\sqrt{\frac{(s-a)(s-b)}{s(s-c)}}$

We get
$cot\frac{A}{2}cot\frac{C}{2}=\sqrt{\frac{s(s-a)}{(s-b)(s-c)}}.\sqrt{\frac{s(s-c)}{(s-a)(s-b)}}=\frac{s}{s-b}=\frac{2s}{2s-2b}$

Now as $a,b,c$ is in A.P

$2b=a+c\Rightarrow 3b=a+b+c=2s$ , we get

$cot\frac{A}{2}cot\frac{C}{2}=\frac{3b}{3b-2b}=3$