Interesting Integrals 10

Calculus Level 4

\[\large\int_{0}^{\sin^2 (x)}{\sin^{-1} \left( \sqrt t \right)}\,dt\ + \large\int_{0}^{\cos^2 (x)}{\cos^{-1} \left( \sqrt t \right)}\,dt=\dfrac{\pi}{c} \]

The equation above holds true for some positive integer \(c\). Find \(c\).