Question

If \[ \tan\beta = 2\sin\alpha \sin\gamma \cdot \csc(\alpha+\gamma), \] then \(\cot\alpha,\; \cot\beta,\; \cot\gamma\) are in:

• A. A.P.
• B. G.P.
• C. H.P.
• D. none of these

Hint:

If \[ 2b=a+c, \] then \(a,b,c\) are in A.P. Look for such mean-value relations when dealing with trigonometric progressions.

Answer 1

The Correct Option is A

Step 1: Simplify the given expression. \[ \tan\beta =2\sin\alpha\sin\gamma\cdot\csc(\alpha+\gamma) =\frac{2\sin\alpha\sin\gamma}{\sin(\alpha+\gamma)} \] Using the identity: \[ \sin(\alpha+\gamma)=\sin\alpha\cos\gamma+\cos\alpha\sin\gamma \]
Step 2: Express in terms of cotangents. Divide numerator and denominator by \(\sin\alpha\sin\gamma\): \[ \tan\beta =\frac{2}{\cot\alpha+\cot\gamma} \]
Step 3: Take reciprocal to obtain \(\cot\beta\). \[ \cot\beta=\frac{\cot\alpha+\cot\gamma}{2} \]
Step 4: Interpret the result. The above relation implies: \[ 2\cot\beta=\cot\alpha+\cot\gamma \] Hence, \(\cot\beta\) is the arithmetic mean of \(\cot\alpha\) and \(\cot\gamma\). Therefore, \(\cot\alpha,\;\cot\beta,\;\cot\gamma\) are in arithmetic progression.
Hence, the correct answer is \[ \boxed{\text{A.P.}} \]