Let a, b, c be the sides of any triangle ABC. Then by applying the sine rule, we get

⇒ a = k sin A, b = k sin B, c = k sin C

So by considering the RHS of the given equation, we get

RHS = c cos(A - θ) + a cos(C + θ)

Substituting the corresponding values from sine rule, we get

⇒ = k sin C cos(A - θ) + k sin A cos(C + θ)

By cancelling like terms we get

⇒ = k sin B cos θ

⇒ = b cos θ (from sine rule b = k sin B)

= LHS

Hence proved