How do you verify #(sin x)(tan x cos x - cot x cos x) = 1 - 2 cos2x#?

2 Answers
May 1, 2018

Pretty sure the question is #(sinx)(tanxcosx-cotxcos x)=1-2cos^2x# ,or else it will be not provable.

Explanation:

Some basic knowledge to begin with:
1. #sin^2x+cos^2x=1#
2. #tanx=sinx/cosx#
3. #cotx=cosx/sinx#

Let's start from the left hand side
#(sinx)(tanxcosx-cotxcos x)#

#=sinxtanxcosx-sinxcotxcosx#

#=sinx(sinx/cosx)cosx-sinx(cosx/sinx)cosx#

#=sin^2x-cos^2x#

#=sin^2x+cos^2x-2cos^2x#

#=1-2cos^2x#

May 1, 2018

#"see explanation"#

Explanation:

#"using the "color(blue)"trigonometric identities"#

#•color(white)(x)tanx=sinx/cosx" and "cotx=cosx/sinx#

#•color(white)(x)sin^2x+cos^2x=1#

#rArrsin^2x=1-cos^2x#

#"consider the left side"#

#"distributing the parenthesis"#

#sinx xxsinx/cancel(cosx)xxcancel(cosx)-cancel(sinx)xxcosx/cancel(sinx)xxcosx#

#=sin^2x-cos^2x#

#=(1-cos^2x)-cos^2x#

#=1-2cos^2x=" right side "rArr"verified"#