Question

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

Answer 1

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)`