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