Question

If \[ \left| 1 - \cos \left( \frac{\pi}{2} - \alpha \right) + \sin \left( \frac{3\pi}{2} - \alpha \right) \right| \left[ 1 - \sin \left( \frac{3\pi}{2} - \alpha \right) - \cos \left( \frac{\pi}{2} - \alpha \right) \right] = a + b \sin \left( \frac{\pi}{4} + \alpha \right) \] then \( a^2 + b^2 = \)

• A. 20
• B. 52
• C. 40
• D. 32

Hint:

Simplify complex trigonometric expressions by using known identities and break down the components carefully.

Answer 1

The Correct Option is B

We are given the equation: \[ \left| 1 - \cos \left( \frac{\pi}{2} - \alpha \right) + \sin \left( \frac{3\pi}{2} - \alpha \right) \right| \left[ 1 - \sin \left( \frac{3\pi}{2} - \alpha \right) - \cos \left( \frac{\pi}{2} - \alpha \right) \right] = a + b \sin \left( \frac{\pi}{4} + \alpha \right) \] Step 1: Simplify the trigonometric expressions Start with the following trigonometric identities: - \( \cos \left( \frac{\pi}{2} - \alpha \right) = \sin \alpha \) - \( \sin \left( \frac{3\pi}{2} - \alpha \right) = -\cos \alpha \) - \( \sin \left( \frac{3\pi}{2} - \alpha \right) = -\cos \alpha \) - \( \cos \left( \frac{\pi}{2} - \alpha \right) = \sin \alpha \) Substitute these into the given expression: \[ \left| 1 - \sin \alpha - \cos \alpha \right| \left[ 1 + \cos \alpha - \sin \alpha \right] \] Step 2: Expand and simplify Expand the expression: \[ \left( 1 - \sin \alpha - \cos \alpha \right) \left( 1 + \cos \alpha - \sin \alpha \right) \] Simplify this: \[ (1 - \sin \alpha - \cos \alpha)(1 + \cos \alpha - \sin \alpha) = (a + b \sin \left( \frac{\pi}{4} + \alpha \right)) \] We can then solve for \( a^2 + b^2 = 52 \). Thus, the correct answer is option (2).