Question

If \( \cos\theta \), \( \sin\theta \), and \( \cot\theta \) are in geometric progression, then: \[ \sin^9\theta + \sin^6\theta + 3\sin^5\theta + \sin^3\theta + \sin^2\theta = \]

• A. \( 2 \)
• B. \( 7 \)
• C. \( 1 \)
• D. \( 5 \)

Hint:

For sums involving trigonometric functions in geometric progression, simplify the terms using the geometric progression relationship and use symmetry.

Answer 1

The Correct Option is A

Step 1: Use geometric progression properties.
Given that \( \cos\theta \), \( \sin\theta \), and \( \cot\theta \) are in geometric progression, this means: \[ \frac{\sin\theta}{\cos\theta} = \frac{\cot\theta}{\sin\theta} \] which simplifies to: \[ \sin^2\theta = \cos\theta \cot\theta \]
Step 2: Apply the geometric progression to the sum.
After applying the properties and solving the sum, we get: \[ \sin^9\theta + \sin^6\theta + 3\sin^5\theta + \sin^3\theta + \sin^2\theta = 2 \]