Question

for any real number X and Y, prove that 1) cos(x+y) =cosXcosY-sinXsinY 2) cos(x-y) =cosXcosY +sinXsinY​

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.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