Prove and apply the identities for products of sines and cosines expressed as sums and differences. sin(𝐴) sin(𝐵) = 1 2 (cos(𝐴 − 𝐵) − cos(𝐴 + 𝐵)) cos(𝐴) cos(𝐵) = 1 2 (cos(𝐴 − 𝐵) + cos(𝐴 + 𝐵)) sin(𝐴) cos(𝐵) = 1 2 (sin(𝐴 + 𝐵) + sin(𝐴 − 𝐵)) cos(𝐴) sin(𝐵) = 1 2 (sin(𝐴 + 𝐵) − sin(𝐴 − 𝐵))

Use the product-to-sum identity to express \(\cos(5x) \cos(2x)\) as a sum of cosines, then evaluate the expression when \(x = \frac{\pi}{6}\).

Use the angle sum and difference identities for sine to prove the product-to-sum identity \(\sin(A) \cos(B) = \frac{1}{2}(\sin(A + B) + \sin(A - B))\).

Using the compound angle formulae provided in the stimulus, prove that \(\cos(A) \cos(B) = \frac{1}{2}(\cos(A - B) + \cos(A + B))\) for \(A, B \in \mathbb{R}\). Hence, use this identity to evaluate \(\cos\left(\frac{7\pi}{12}\right) \cos\left(\frac{\pi}{12}\right)\) exactly.

Prove the identity \(\sin(A)\cos(B) = \frac{1}{2}(\sin(A+B) + \sin(A-B))\).