Question

If \( 1 + \sqrt{1 + a} = (1 + \sqrt{1 - a}) \cot \alpha \) and \( 0<a<1 \), then \( \sin 4\alpha = \)

• A. \( a \)
• B. \( 2a \)
• C. \( 3a \)
• D. \( 4a \)

Hint:

When working with trigonometric identities, solve for the base angle first, then use known identities for higher multiples of the angle.

Answer 1

The Correct Option is A

We are given the equation: \[ 1 + \sqrt{1 + a} = (1 + \sqrt{1 - a}) \cot \alpha \] First, solve for \( \cot \alpha \): \[ \cot \alpha = \frac{1 + \sqrt{1 + a}}{1 + \sqrt{1 - a}} \] Using the identity \( \cot^2 \alpha = \csc^2 \alpha - 1 \), we can substitute into the formula for \( \sin 4\alpha \) using the standard identity for the sine of a multiple angle: \[ \sin 4\alpha = 4 \sin \alpha \cos \alpha \] After solving for \( \sin \alpha \) and \( \cos \alpha \) from the given equation, we find that: \[ \sin 4\alpha = a \] Thus, the correct answer is \( a \), which corresponds to option (1).