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Question:

Prove the following: cos(π/4 - x)cos(π/4 - y) - sin(π/4 - x)sin(π/4 - y) = sin(x + y)

Solution:

LHS = cos(π/4 - x)cos(π/4 - y) - sin(π/4 - x)sin(π/4 - y)
= cos(π/4 - x + π/4 - y) Using identity cosAcosB - sinAsinB = cos(A + B)
= cos(π/2 - (x + y))
= sin(x + y)
= RHS
∴ cos(π/4 - x)cos(π/4 - y) - sin(π/4 - x)sin(π/4 - y) = sin(x + y)
Hence proved