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Question

cos(3 pie /4+x)-cos((3 pie /4-x)=-root2 sinx

Answer 1

Question: Prove the identity: cos(3 pie /4+x)-cos((3 pie /4-x)=-root2 sinx.

expand the cosines.

lhs = cos(3 pie /4+x) - cos(3 pie /4-x)

lhs = cos(3π/4).cos(x) - sin(3π/4).sin(x) - {cos(3π/4).cos(x) + sin(3π/4).sin(x) }

lhs = -2.sin(3π/4).sin(x)

and sin(3π/4) = 1/√2, hence

lhs = -√2.sin(x)

But rhs = -√2.sin(x)

i.e. lhs = rhs, for all x.

Therefore the identity is proven