Let $\displaystyle I=\int_{0}^{\frac{\pi}{2}} \frac{\sin^{50}x}{\sin^{50}x+\cos^{50}x}dx\qquad \to (1)$
By using property of definite integration
$\displaystyle \int_{0}^{a}f(x)dx=\int_{0}^{a}f(a-x)dx$
$\displaystyle I=\int_{0}^{\frac{\pi}{2}} \frac{\cos^{50}x}{\cos^{50}x+\sin^{50}x}dx\qquad \to (2)$
Adding $(1)$ and $(2),$
$\displaystyle 2I=\int_{0}^{\frac{\pi}{2}} \frac{\sin^{50}x+\cos^{50}x}{\sin^{50}x+\cos^{50}x}dx$
$\displaystyle \implies 2I=\int_{0}^{\frac{\pi}{2}}1\;dx$
$\displaystyle \implies 2I=\left[x\right]_{0}^{\frac{\pi}{2}}$
$\implies 2I=\left[ \frac{\pi}{2}\right]$
$\implies I=\frac{\pi}{4}$
So, answer is (C).