How do you prove cosx- (cosx/(1-tanx))= (sinxcosx)/(sinx-cosx)?

1 Answer
Feb 17, 2016

Please see below for the proof. Feel free to ask questions if you have any.

Explanation:

1) replace tanx with sinx/cosx

cosx-(cosx/(1-tanx))
= cosx - (cosx/(1-(sinx/cosx)))

2) equalise the denominator in the paranthesis

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

=cosx - (cosx*cosx/(cosx-sinx))
=cosx-cos^2x/(cosx-sinx)

3) equalise the denominator once more
=cosx(cosx-sinx) /(cosx-sinx)- cos^2x/(cosx-sinx)
=(cos^2x-cosx*sinx-cos^2x)/(cosx-sinx)
=(-(cosx*sinx))/(cosx-sinx)

4) put the denominator into -1 paranthesis
=-(cosx*sinx)/(-(-cosx+sinx))

5) - divided by - yields +. And change the places of cosx and sinx in the denominator

=(cosx*sinx)/(sinx-cosx)

=(sinx*cosx)/(sinx-cosx)