If for the solution curve \( y=f(x) \) of the differential equation \(\displaystyle \frac{dy}{dx}+(\tan x) y=\frac{2+\sec x}{(1+2\sec x)^{2}} \), \(\displaystyle x\in\left(\frac{-\pi}{2},\frac{\pi}{2}\right) \), \(\displaystyle f\!\left(\frac{\pi}{3}\right)=\frac{\sqrt{3}}{10} \), then \(\displaystyle f\!\left(\frac{\pi}{4}\right) \) is equal to:
