Question

In \(\Delta ABC\), with usual notations, if \(1-\tan \frac{A}{2}\cdot \tan \frac{C}{2}=\frac{b}{k}\), then \(k=\) ...

• A. \(s\)
• B. \(2s\)
• C. \((s-a)\)
• D. \((s-a)(s-b)\)

Hint:

Use tan(A/2)tan(C/2) = (s-b)/s and simplify.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
Half-angle formulae: \(\tan\dfrac A2=\sqrt{\dfrac{(s-b)(s-c)}{s(s-a)}}\) and \(\tan\dfrac C2=\sqrt{\dfrac{(s-a)(s-b)}{s(s-c)}}\).

Step 2: Multiply
\(\tan\dfrac A2\tan\dfrac C2=\sqrt{\dfrac{(s-b)^{2}}{s^{2}}}=\dfrac{s-b}{s}\)

Step 3: Simplify
\(1-\tan\dfrac A2\tan\dfrac C2=1-\dfrac{s-b}{s}=\dfrac bs\)

Step 4: Compare
Given that this equals \(\dfrac bk\), we get \(k=s\), option (A).

Final Answer:
The value of k is s, the semi-perimeter. \[ \boxed{\text{(A) }s} \]