Replace $y$ by $y+t\delta y$ and then compute

$$\lim_{t\to 0} \Bigl(\; F(y+t\delta y)-F(y)\;\Bigr). $$

Here are the details. Observe first that

$$\frac{1}{y+ A+ t\delta y}-\frac{1}{y+A} = -\frac{1}{(y+A)^2} t\delta y+ O(t^2),$$

so that

$$F(y+t\delta y)= \left(\int\;\Bigl(\; \frac{1}{y+A} -\frac{1}{(y+A)^2} t\delta y+O(t^2)\;\Bigr) \cos(x) dx\right)^2 $$

$$= F(y) -2t\left(\int\frac{\cos (x)}{y+A}\right)\left(\int\frac{(\delta y)\cos(x)}{\bigl(y+A)^2} dx\right) +O(t^2). $$

From here you can read that

$$\frac{\delta F}{\delta y} =-2 \left(\int\frac{\cos (x)}{y+A}dx\right)\frac{\cos(x)}{\bigl(\;y(x)+A \;\bigr)^2} $$