Question

prove that cos(x+y)=cosx*cosy-sinx*siny​

Answer 1

Let us take a circle of radius one and let us take 2 points P and Q such that P is at an angle x and Q at an angle y

as shown in the diagram

Therefore, the co-ordinates of P and Q are P(cosx,sinx),Q(cosy,siny)

Now the distance between P and Q is:

(PQ) 2 =(cosx−cosy) 2 +(sinx−siny) 2=2−2(cosx.cosx+sinx.siny)

Now the distance between P and Q u\sin g \cos ine formula is

(PQ) 2 =1 2 +1 2 −2cos

(x−y)=2−2cos(x−y)

Comparing both we get

cos(x−y)=cos(x)cos(y)+sin(x)sin(y)

Substituting y with −y we get

cos(x+y)=cosxcosy−sinxsiny

Hope this'll help you.