If ∫dxsin4⁡x+cos4⁡x=12tan−1⁡f(x)+C then

Dividing the numerator and denominator by cos4⁡x, the given integral can be written as ∫sec2⁡x1+tan2⁡xtan4⁡x+1dx=∫1+t21+t4dt(t=tan⁡x)=∫1+1/t2t2+1/t2dt=∫1+1/t2t−1t2+2dt=12tan−1⁡12t−1t+C=12tan−1⁡tan⁡x−cot⁡x2+CThus f(x)=tan⁡x−cot⁡x2 so fπ4=0