Question

If \( \tan^{-1} (\sqrt{\cos \alpha}) - \cot^{-1 (\cos \alpha) = x \), then what is \( \sin \alpha \)?}

• A. \( \tan \left( \frac{x}{2} \right) \)
• B. \( \cot \left( \frac{x}{2} \right) \)
• C. \( \cot^2 \left( \frac{x}{2} \right) \)
• D. \( \tan^2 \left( \frac{x}{2} \right) \)

Hint:

Use sum and difference identities for inverse trigonometric functions to simplify complex equations.