Using double angle formula, we know that $$cos(2 heta) = cos^2 heta - sin^2 heta$$ => $$cos(2 heta) = (1 - sin^2 heta) - sin^2 heta$$ => $$cos(2 heta) = 1 - 2sin^2 heta$$ Replacing $$ heta$$ by $$frac{A}{2}$$, we get : => $$cos A = 1 - 2sin^2(frac{A}{2})$$ => $$2sin^2(frac{A}{2}) = 1 - cosA$$ => $$sin^2(frac{A}{2}) = frac{(1-cosA)}{2}$$ => $$sin(frac{A}{2}) = sqrt{frac{(1 - cos A)}{2}}$$ Similarly, => $$cos(frac{A}{2}) = sqrt{frac{(1 + cos A)}{2}}$$ Now, to find : $$cot(frac{A}{2})$$ = $$cos(frac{A}{2}) div sin(frac{A}{2})$$ = $$sqrt{frac{(1 + cos A)}{2}}$$ $$div$$ $$sqrt{frac{(1 - cos A)}{2}}$$ = $$sqrt{frac{(1 + cos A)}{2}}$$ $$ imes$$ $$sqrt{frac{2}{(1 - cos A)}}$$ = $$sqrt{frac{1+cosA}{1-cosA}}$$ = $$sqrt{frac{1+cosA}{1-cosA} imes frac{1-cosA}{1-cosA}}$$ = $$sqrt{frac{1-cos^2A}{(1-cosA)^2}} = sqrt{frac{sin^2A}{(1-cosA)^2}}$$ = $$frac{sinA}{1-cosA}$$ Dividing numerator and denominator by $$(sinA)$$ = $$frac{1}{cosecA - cotA}$$ => Ans - (D)