Question

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

Answer 1

To prove :

cos (x + y) = cosx.cosy-sinx.siny

Proof :

we know,

cos (-x) = cos x ,

sin (-x) = - sin x

and

cos (x - y)

= cos x . cos y + sin x . sin y

therefor,

LHS = cos (x + y) = cos (x - (-y))

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

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

=)) cos x. cos y - sinx . sin y

= RHS

hence proved

Answer 2

Answer:

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