Question

If cos 2B = `(cos(A + C))/(cos(A - C))`, then tan A, tan B, tan C are in ______.

विकल्प

• AP

• GP

• HP

• None of these

Answer 1

If cos 2B = `(cos(A + C))/(cos(A - C))`, then tan A, tan B, tan C are in GP.

Explanation:

cos 2B = `(cos(A + C))/(cos(A - C))`

= `(cos A cos C - sin A sin C)/(cos A cos C + sin A sin C)`

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

`\implies` 1 + tan² B – tan A tan C – tan A tan C tan² B

= 1 – tan² B + tan A tan C – tan A tan C tan² B

`\implies` 2 tan² B = 2 tan A tan C

`\implies` tan² B = tan A tan C

Hence, tan A, tan B and tan C will be in GP.