Question

If \[ \left(\frac{\cos \theta + i \sin \theta}{\sin \theta + i \cos \theta}\right)^{2024} + \left(\frac{1 + \cos \theta + i \sin \theta}{1 - \cos \theta + i \sin \theta}\right)^{2025} = x + iy, \] and $x + y$ at $\theta = \frac{\pi}{2}$ is

• A. 1
• B. -1
• C. 2
• D. 2024

Hint:

Evaluate complex expressions at given angles carefully by substituting trigonometric values and simplifying powers of $i$ using cyclical properties.

Answer 1

The Correct Option is C

Substitute $\theta = \frac{\pi}{2}$: \[ \cos \frac{\pi}{2} = 0, \quad \sin \frac{\pi}{2} = 1. \] Evaluate the first term: \[ \frac{\cos \theta + i \sin \theta}{\sin \theta + i \cos \theta} = \frac{0 + i \times 1}{1 + i \times 0} = \frac{i}{1} = i. \] Thus, \[ % Option (i)^{2024} = (i^4)^{506} = 1^{506} = 1. \] Evaluate the second term: \[ \frac{1 + \cos \theta + i \sin \theta}{1 - \cos \theta + i \sin \theta} = \frac{1 + 0 + i \times 1}{1 - 0 + i \times 1} = \frac{1 + i}{1 + i} = 1. \] So, \[ 1^{2025} = 1. \] Therefore, \[ x + iy = 1 + 1 = 2. \] Since $x + y = 2$, the correct answer is 2.