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Question

Prove that: cos(π4x)cos(π4y)sin(π4x)sin(π4y)=sin(x+y)

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Solution

L.H.S=cos(π4x)cos(π4y)sin(π4x)sin(π4y)
Using formula, cos(A+B)=cos Acos Bsin Asin B
Let A=π4x and B=π4y
cos[(π4x)+(π4y)]
=cos[π4x+π4y]
=cos[π2(x+y)]=sin(x+y){cos(π2θ)=sinθ}
So, LH.S =R.H.S
cos(π4x)cos(π4y)sin(π4x)sin(π4y)=sin(x+y)
Hence proved.

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