Xtan2x = 2 tang -equals: *** (1-cos...
Question

# Xtan2x = 2 tang -equals: *** (1-cos2x) (2018 (b) IN

JEE/Engineering Exams
Maths
Solution
128
4.0 (1 ratings)
( x tan 2 x-2 x tan x ) ( lim _{x rightarrow 0} frac{1}{x} ) ( (1-cos 2 x) ) ( lim _{x rightarrow 0} frac{x[tan 2 x-2 tan x]}{(1-cos 2 x)^{2}} ) ( lim _{x rightarrow 0} frac{xleft[frac{2 tan x}{1+tan ^{2} x}-2 tan xright.}{sin ^{2} x^{4}+x} ) ( =lim _{x rightarrow 0} frac{xleft[2 tan x-2 tan x+2 tan ^{3} xright]}{4 sin ^{2} x}left(1-tan ^{2} xright) ) ( =lim _{x rightarrow 0} frac{2 x tan ^{3} x}{4 sin ^{4} xleft(1-tan ^{2} xright)} ) ( =lim _{x rightarrow 0} frac{2 x sin ^{3} x}{2 sin ^{4} x cos ^{3} xleft(1-tan ^{2} xright)} ) ( =lim _{x rightarrow 0} frac{x}{2 sin x cos ^{3} xleft(1-frac{sin ^{2} x}{cos ^{2} x}right)} ) ( =lim _{x rightarrow 0} frac{x operatorname{los}^{2} x}{2 sin x cos ^{3} xleft(cos ^{2} x-sin ^{2} xright)} ) ( =lim _{x rightarrow 0} frac{x}{2 sin x cos xleft(cos ^{2} x-sin ^{2} xright)} ) ( begin{aligned} therefore sin 2 theta &=2 sin theta cos theta ln cos 2 theta &=cos ^{2} theta-sin ^{2} theta end{aligned} ) ( =lim _{n rightarrow 0} frac{2 x}{2 sin 2 x cos 2 x} ) ( =lim _{x rightarrow 0} frac{2 x}{sin 4 x} ) ( frac{0}{0} ) form - Abply L'twoitals rule ( (c) )