Prove that 1-cos5Acos3A-sin5Asin3A=2sin^2A  Please give every steps if possible for proving it.... :)

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We'll re-write the expression:

1 - (cos 5Acos3A + sin5Asin3A) = 2`sin^2 A`

We'll use the identity:

cos(a-b) = cos a*cos b + sin a*sin b

(cos 5Acos3A + sin5Asin3A) = cos (5A-3A)

(cos 5Acos3A + sin5Asin3A) = cos 2A

1 - cos 2A = `2sin^2A`

cos 2A = 1 - `2sin^2 A`

Since we've get the double angle identity, that means that the given expression represents an identity.

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