Question

Prove geometrically that cos(x - y) = cos x cos y + sin x sin y.​

Answer 1

Answer:

Step-by-step explanation:

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.cosy+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