Walli's Formula

Powers of sine or cosine integrated over the quarter period [0, \pi/2] are governed by Walli's formula, which compresses an otherwise repetitive integration-by-parts recursion into a single ratio. For an even exponent n the formula reads \int_0^{\pi/2}\cos^n x,dx = \frac{(n-1)(n-3)\cdots 1}{n(n-2)\cdots 2}\cdot\frac{\pi}{2}, where the extra \pi/2 factor appears only for even powers. For n = 4 this gives \frac{3\cdot 1}{4\cdot 2}\cdot\frac{\pi}{2} = \frac{3}{8}\cdot\frac{\pi}{2} = \frac{3\pi}{16}. As an independent confirmation, writing \cos^4 x = \left(\frac{1+\cos 2x}{2}\right)^2 and expanding into constant and double-angle terms reproduces the same value after integration. Option \pi/4 corresponds to \cos^2 x, which is one power too few. Option \pi/2 is the integral of 1, ignoring the fourth power entirely. Option 3\pi/8 omits the final \pi/2 factor that even powers demand. The formula neatly systematizes this whole family of symmetric power integrals. As a final plausibility check, since \cos^4 x \le 1 everywhere on the interval, the integral must stay strictly below the integral of one over the same range, which is \pi/2 \approx 1.571, and the computed value 3\pi/16 \approx 0.589 sits comfortably within that upper bound, exactly as a fourth power averaging well under one would predict.