write the given product as a sum. You may need to use an Even/Odd Identity. cos(2θ) cos(6θ)

Question

i have the answer and the answer is cos(4θ)+cos(8θ) /2 and they are both over 2 i just have no clue how to show my work any help would be amazing thx

Answer 1

cos(A + B) = cos(A)cos(B) - sin(A)sin(A)

cos(A - B) = cos(A)cos(B) + sin(A)sin(A)

Adding equations:

(1/2)(cos(A + B) + cos(A - B)) = cos(a)cos(A)

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cos(a)cos(A) = (1/2)(cos(A + B) + cos(A - B))

cos(2θ)cos(6θ) = (1/2)(cos(8θ) + cos(-4θ))

Note that cos(-4θ) = cos(4θ) ← even identity

Therefore,

cos(2θ)cos(6θ) = (1/2)cos(8θ) + (1/2)cos(4θ)