How do you verify #cotx-tanx=2cot2x#?

2 Answers
Jul 24, 2015

Use the double angle formula for #tan(2x)# and the fact that #cot(theta) = 1/tan(theta)#

Explanation:

Double angle formula for #tan#
#color(white)("XXXX")##tan(2x) = (2tan(x))/(1-tan^2(x))#

#2cot(2x) = 2* 1/tan(2x)#

#color(white)("XXXX")##=2* (1-tan^2(x))/(2 tan(x))#

#color(white)("XXXX")##=1/tan(x) - tan^2(x)/tan(x)#

#color(white)("XXXX")##=cot(x) - tan(x)#

Apr 6, 2017

Using the relationship between tan/cot and sin-cos, plus the double angle formulae for sin and cos. (as requested)

Explanation:

Remember:

#color(red)("Basic definitions:")#
#color(white)("XXX")color(red)(tan(theta)=sin(theta)/cos(theta)color(white)("XXX")cot(theta)=cos(theta)/sin(theta))#

#color(blue)("Double angle formulae for sin and cos")#
#color(white)("XX"color(blue)(sin(2theta)=2 * sin(theta) * cos(theta)color(white)("XX")cos(2theta)=cos^2(theta)-sin^2(theta))#

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Required to Prove:

#color(green)(cot(x)-tan(x)=2cot(2x)#

Proof:

#R.S.#
#color(white)("XXX")=color(green)(2cot(2x))#

#color(white)("XXX")=2 * cos(2x)/sin(2x)#

#color(white)("XXX")=(cancel2 * (cos^2(x)-sin^2(x)))/(cancel2 * sin(x) * cos(x))#

#color(white)("XXX")=cos^2(x)/(sin(x) * cos(x)) - sin^2(x)/(sin(x) * cos(x))#

#color(white)("XXX")=cos(x)/sin(x) -sin(x)/cos(x)#

#color(white)("XXX")=color(green)(cot(x)-tan(x))#

#color(white)("XXX")=L.S.#