(A) tan A, tan C, tan B are in A.P.

(B) tan A, tan C, tan B are in G.P.

(C) tan A, tan B, tan C are in G.P.

(D) tan A, tan B, tan C are in A.P.

Correct Answer: tan A, tan C, tan B are in G.P.

Explanation

Given,

tan(A − B) = (tan A − tan B)/(1 + tan A tan B)

Substitute in the given expression:

(tan A − tan B)/(tan A(1 + tan A tan B)) + sin²C/sin²A = 1

Using

sin²C/sin²A = tan C/tan A

After simplification, we get:

(tan C)² = tan A · tan B

This implies tan A, tan C, tan B are in geometric progression.

Hence, tan A, tan C, tan B are in G.P.

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