given : sin (x+y) = sin x cos y + cos x sin y and cos (x+y) = cos x cos y-sin x sin y derive a formula for cot (x+y) and derive a formula for cos(2x)

Question

given : sin (x+y) = sin xcos y + cos x sin y

and cos(x+y) = cos x cos y - sin x sin y

Answer 1

cot(x+y)=cos(x+y)/sin(x+y)=(cosxcosy-sinxsiny)/(sinxcosy+cosxsiny).

There are variations of this: divide top and bottom by cosx: (cosy-tanxsiny)/(tanxcosy+siny),

then by cosy: (1-tanxtany)/(tanx+tany). Dividing by sinx and siny instead gives:

(cotxcosy-siny)/(cosy+cotxsiny), then: (cotxcoty-1)/(coty+cotx).

(Using these variations and putting y=x, you can work out cot(2x).)

cos(2x)=cos^2(x)-sin^2(x), putting y=x.

This can also be written: 1-2sin^2(x) or 2cos^2(x)-1 because sin^2(x)+cos^2(x).