If A + B + C = 180°, then the value of cot A/2 + cot B/2 + cot C/2 will be

1) 2 cot A/2 cot B/2 cot C/2

2) 4 cot A/2 cot B/2 cot C/2

3) cot A/2 cot B/2 cot C/2

4) 8 cot A/2 cot B/2 cot C/2

Answer: (3) cot A/2 cot B/2 cot C/2

Solution:

Given,

A + B + C = 180°

Dividing by 2 on both sides,

A/2 + B/2 + C/2 = 180°/2

A/2 + B/2 = 90° – C/2

Taking “cot” on both sides,

cot(A/2 + B/2) = cot(90° – C/2)

[cot A/2 cot B/2 – 1]/[cot A/2 + cot B/2] = tan C/2

[cot A/2 cot B/2 – 1]/[cot A/2 + cot B/2] = 1/[cot C/2]

cot C/2[cot A/2 cot B/2 – 1] = cot A/2 + cot B/2

cot A/2 cot B/2 cot C/2 – cot C/2 = cot A/2 + cot B/2

Therefore, cot A/2 + cot B/2 + cot C/2 = cot A/2 cot B/2 cot C/2

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