Question

Let A, B and C are the angles of a triangle and `tan(A/2) = 1/3, tan(B/2) = 2/3`. Then, `tan(C/2)` is equal to ______.

Options

• `1/3`

• `2/3`

• `2/9`

• `7/9`

Answer 1

Let A, B and C are the angles of a triangle and `tan(A/2) = 1/3, tan(B/2) = 2/3`. Then, `tan(C/2)` is equal to `underlinebb(7/9)`.

Explanation:

Given, A + B + C = π

`\implies (A + B)/2 = π/2 - C/2`

`\implies tan((A + B)/2) = tan(π/2 - C/2) = cot  C/2`

`\implies (tan  A/2 + tan  B/2)/(1 - tan  A/2 tan  B/2) = cot  C/2`  

`\implies (1/3 + 2/3)/(1 - 1/3 xx 2/3) = cot  C/2`  ...`[∵ tan  A/2 = 1/3, tan  B/2 = 2/3 ("given")]`

`\implies cot  C/2 = 9/7`

`\implies tan  C/2 = 7/9`