Question

If \(\cos^ 4\theta - \sin^ 4\theta = \frac{1}{2}\),then \(θ=\)

• A. 30°
• B. 60°
• C. 45°
• D. None

Answer 1

The Correct Option is A

We are given: \[ \cos^4 \theta - \sin^4 \theta = \frac{1}{2} \] Using the difference of squares formula: \[ \cos^4 \theta - \sin^4 \theta = (\cos^2 \theta - \sin^2 \theta)(\cos^2 \theta + \sin^2 \theta) \] Since \(\cos^2 \theta + \sin^2 \theta = 1\), the equation simplifies to: \[ \cos^2 \theta - \sin^2 \theta = \frac{1}{2} \] This is the standard identity for \(\cos(2\theta)\): \[ \cos(2\theta) = \frac{1}{2} \] From trigonometric identities, we know: \[ \cos(60^\circ) = \frac{1}{2} \] Thus, \(2\theta = 60^\circ\), which gives: \[ \theta = 30^\circ \]

The correct answer is option (A): \(30°\)