Prove the identity cos (x-y) - cos (x+y) = 2sinxsiny

Question

Mathematics

Elias Tanner

12 August, 17:02

Prove the identity cos (x-y) - cos (x+y) = 2sinxsiny

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Preston Hayes · Answer 1

cos (x-y) - cos (x+y) = 2sinxsiny

Use cosine addition and subtraction identities:

cos (x-y) = cos x cos y + sin x sin y

cos (x+y) = cos x cos y - sin x sin y

So

LHS = cos (x-y) - cos (x+y)

= cos x cos y + sin x sin y - (cos x cos y - sin x sin y)

= cos x cos y + sin x sin y - cos x cos y + sin x sin y

cos x cos y and - cos x cos y are additive inverses; they sum together to get 0.

LHS = sin x sin y + sin x sin y

= 2 sin x sin y

= RHS