Question

In \(\triangle ABC\), if \(a + c = 5b\), then \(\cot\dfrac{A}{2} \cdot \cot\dfrac{C}{2} =\)

• A. 2
• B. \(\dfrac{1}{2}\)
• C. \(\dfrac{3}{2}\)
• D. \(\dfrac{2}{3}\)

Hint:

Use the cotangent-half angle identity for triangles and simplify using triangle relations or constraints.

Answer 1

The Correct Option is C

Using cotangent half-angle identities: \[ \cot\dfrac{A}{2} = \sqrt{\dfrac{(s-b)(s-c)}{s(s-a)}},\quad \cot\dfrac{C}{2} = \sqrt{\dfrac{(s-a)(s-b)}{s(s-c)}}. \] Their product simplifies using the condition \(a + c = 5b\), leading to \(\dfrac{3}{2}\).