hi again,firstly many thanks for the very prompt replies.

i'm sorry to say i not quite getting there which is annoying me.

i know the lhs needs to equal the rhs and that the identities can represent other identities and thats how it is all simplified but i cant seem to convert to get the appropriate identities and the theory is so simple!

my ongoing attempt, is this getting there or not?

cos2x.cosx-sin2x.sinx changes to the following?

2cos^2-1.cosx-sin2x.sinx

2cos^3x-1-sinx.sinx

2cos^3-1-sin^2 2x?

so its all going a bit pete tong as they say

not seeing how this will equate to the rhs i'm afraid and i have honestly been trying for hours now.