Question

In a triangle \(ABC\) with usual notations, if \(\tan A,\tan B,\tan C\) are in H.P., then \(a^2,b^2,c^2\) are in

• A. A.P.
• B. Not in A.P.
• C. H.P.
• D. G.P.

Hint:

In triangle problems, harmonic progression of trigonometric ratios often leads to arithmetic progression among squares of sides.

Answer 1

The Correct Option is A

Step 1: Use identity in a triangle.
In any triangle, \[ \tan A+\tan B+\tan C=\tan A\tan B\tan C \] Step 2: Property of H.P.
If \(\tan A,\tan B,\tan C\) are in H.P., then their reciprocals are in A.P.
Step 3: Relate sides and angles.
Using standard triangle identities, it follows that \[ a^2,b^2,c^2 \] are in arithmetic progression.
Step 4: Conclusion.
Hence, the correct answer is A.P.