Question:

Let $I = \int_{0}^{\pi/2} \frac{g(\sin x)}{g(\sin x) + g(\cos x)} dx \Rightarrow I = \int_{0}^{\pi/2} \frac{g\left(\sin\left(\frac{\pi}{2}-x\right)\right)}{g\left(\sin\left(\frac{\pi}{2}-x\right)\right) + g\left(\cos\left(\frac{\pi}{2}-x\right)\right)} dx$<br>$= \int_{0}^{\pi/2} \frac{g(\cos x)}{g(\cos x) + g(\sin x)} dx$<br>on adding, we obtain<br>$2I = \int_{0}^{\pi/2} \left( \frac{g(\sin x)}{g(\sin x) + g(\cos x)} + \frac{g(\cos x)}{g(\cos x) + g(\sin x)} \right) dx = \int_{0}^{\pi/2} dx \Rightarrow I = \frac{\pi}{4}$