Answer
See below.
Work Step by Step
$LHS=\frac{tan(u)-cot(u)}{tan(u)+cot(u)}+2cos^2(u)=\frac{tan(u)-cot(u)}{tan(u)+cot(u)}\times\frac{sin(u)cos(u)}{sin(u)cos(u)}+2cos^2(u)=\frac{sin^2(u)-cos^2(u)}{sin^2(u)+cos^2(u)}+2cos^2(u)=sin^2(u)-cos^2(u)+2cos^2(u)=1=RHS$