Euler's Beta and Gamma function relation for trigonometric integrals:
The standard formula for definite integrals involving sines and cosines is:
$\int_0^{\pi/2} \sin^p x \cos^q x \, dx = \frac{\Gamma\left(\frac{p+1}{2}\right) \Gamma\left(\frac{q+1}{2}\right)}{2 \Gamma\left(\frac{p+q+2}{2}\right)}$.
Setting $p = 0$ and $q = n$:
$\int_0^{\pi/2} \cos^n x \, dx = \frac{\Gamma\left(\frac{1}{2}\right) \Gamma\left(\frac{n+1}{2}\right)}{2 \Gamma\left(\frac{n+2}{2}\right)}$.
Since $\Gamma(1/2) = \sqrt{\pi}$:
$\int_0^{\pi/2} \cos^n x \, dx = \frac{\Gamma\left(\frac{n+1}{2}\right)}{\Gamma\left(\frac{n+2}{2}\right)} \frac{\sqrt{\pi}}{2}$.

Option breakdown:

• $\frac{\Gamma(\frac{n+1}{2})}{\Gamma(\frac{n+2}{2})} \frac{\sqrt{\pi}}{2}$ is the standard reduction formula in terms of the Gamma function.
• Inverts the ratio of gamma arguments.
• Has argument $n/2$ instead of $(n+1)/2$ in the numerator.
• Has argument $n/2$ in the denominator instead of $(n+2)/2$.