Question

If \[ \cos^{-1} x - \frac{\cos^{-1} y}{2} = \alpha, \] then \[ 4x^2 - 4xy \cos \alpha + y^2 \] is equal to:

• A. \(2\sin 2\alpha\)
• B. \(4\)
• C. \(4\sin^2\alpha\)
• D. 4-4sin²α

Hint:

Use trigonometric identities to reduce algebraic expressions.

Answer 1

The Correct Option is C

Using the cosine difference identity and simplifying:

\[ 4x^2 - 4xy \cos \alpha + y^2 = (2x - y)^2 + 4xy \sin^2 \alpha \]

Hence, the expression reduces to:
\[ 4 \sin^2 \alpha \]