Formula used:sin ?/cos ? = tan ? cos ?/sin ? = cot ? sin2? + cos2? = 12sin ? cos ? = sin 2? Calculation:\(f(x) = \frac{1}{\tan x+\cot x},\) \(0 < x < \frac{\pi}{2}?\)? \(f(x) = \frac{1}{\tan x+\cot x},\)? \(f(x) = \frac{1}{\frac{sin x}{cosx}+\frac{cosx}{sinx}}\)? \(f(x) =\frac{sinx\ cosx}{sin^2x+cos^2x}\)? f(x) = sin x cos x (\(\frac{2}{2}\)) [? sin2? + cos2? = 1] ? f(x) = \(\frac{1}{2}\)sin 2x [? 2sin ? cos? = sin 2?]We know that, -1 ? sin ? ? 1 ? -1 ? sin 2x ? 1 ? f(x)max = \(\frac{1}{2}\)(1) = 1/2