Question

P.T cos (x + y) = cos x cos y - sin x sin y.​

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

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 yas shown in the diagram

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 yas shown in the diagramTherefore, the co-ordinates of P and Q are P(cosx,sinx),Q(cosy,siny)

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 yas shown in the diagramTherefore, 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)

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

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 getcos(x−y)=cos(x)cos(y)+sin(x)sin(y)

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 getcos(x−y)=cos(x)cos(y)+sin(x)sin(y)Substituting y with −y we get

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 getcos(x−y)=cos(x)cos(y)+sin(x)sin(y)Substituting y with −y we getcos(x+y)=cosxcosy−sinxsiny