The equation $\int_{-π/4}^{π/4}(a|\sin x|+\frac{b\sin x}{1+\cos x}+c)dx=0$, where a, b, c are constant, gives a relation between:

a and c

$I=\int\limits_{-π/4}^{π/4}(a|\sin x|+\frac{b\sin x}{1+\cos x}+c)dx=0$

$⇒\int\limits_{0}^{π/4}[(a|\sin x|+\frac{b\sin x}{1+\cos x}+c]dx+\int\limits_{0}^{π/4}[(a|\sin (-x)|+\frac{b\sin (-x)}{1+\cos (-x)}+c]dx=0$

$⇒\int\limits_{0}^{π/4}(2a|\sin x|+2c)dx=0⇒2a(\frac{-1}{\sqrt{2}}+1)+2c.\frac{\pi}{4}=0$