Question

Evaluate the expression: \[ \sin^2 76^\circ + \sin^2 16^\circ - \sin 76^\circ \sin 16^\circ \]

• A. 0
• B. \( \frac{1}{4} \)
• C. \( \frac{3}{4} \)
• D. \( \frac{4}{3} \)

Hint:

Use the sum-to-product identities to simplify trigonometric expressions.

Answer 1

The Correct Option is C

Let \[ I = \cos^2 76^\circ + \cos^2 16^\circ - \cos 76^\circ \cos 16^\circ \] We can rewrite \( I \) as: \[ I = \cos^2 (60^\circ + 16^\circ) + \cos^2 16^\circ - \cos (60^\circ + 16^\circ) \cos 16^\circ \] Using the identity \( \cos(A + B) = \cos A \cos B - \sin A \sin B \), we get: \[ I = \left( \cos 60^\circ \cos 16^\circ - \sin 60^\circ \sin 16^\circ \right)^2 + \cos^2 16^\circ - \left( \cos 60^\circ \cos 16^\circ - \sin 60^\circ \sin 16^\circ \right) \cos 16^\circ \] Now, expanding the terms: \[ I = \cos^2 16^\circ \left( 4 \right) + 3 \sin^2 16^\circ \left( 4 \right) - \sqrt{3} \cos 16^\circ \sin 16^\circ \left( 2 \right) + \cos^2 16^\circ - \cos^2 16^\circ \left( 2 \right) + \sqrt{3} \cos 16^\circ \sin 16^\circ \left( 2 \right) \] Simplifying further: \[ I = 3 \cos^2 16^\circ \left( 4 \right) + 3 \sin^2 16^\circ \left( 4 \right) \] Thus, the final result is: \[ I = \frac{3}{4} \]