Irrational Number Identification

A non-terminating and non-repeating decimal represents an irrational number. Option A: $\frac{\pi}{2}(\frac{1}{\pi} + 1) - \frac{\pi}{2} = \frac{1}{2} + \frac{\pi}{2} - \frac{\pi}{2} = \frac{1}{2}$ (Rational). Option B: $\sin^2 1^\circ + \dots + \sin^2 89^\circ = 44.5$ (Rational). Option C: $\sqrt{2}(3\sqrt{2} - 2\sqrt{2}) + \sqrt{3} = 2 + \sqrt{3}$ (Irrational). Option D: $\frac{9}{3} + \frac{22}{7} = 3 + \frac{22}{7} = \frac{43}{7}$ (Rational). Option E: Infinite GP sum $\frac{1}{1 - \pi/4} = \frac{4}{4-\pi}$, so $(4 - \pi) \times \frac{4}{4-\pi} = 4$ (Rational). Hence C is irrational.