5. Prove Geometrically that, cos(x + y) = COS X · cos y - sin xsin y.
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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
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