Convert a product of trigonometric functions into a sum (or difference) Algebra II

Convert the following into a sum or difference of trigonometric functions.

\sin\left(\dfrac{3}{4}x\right) \cdot \cos\left(\dfrac{5\pi}{8}x\right)

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\dfrac{1}{2}\sin\left(\dfrac{6+5\pi}{8}x\right) + \dfrac{1}{2}\sin\left(\dfrac{6-5\pi}{8}x\right)

\dfrac{1}{2}\sin\left(\dfrac{6+5\pi}{8}x\right) - \dfrac{1}{2}\cos\left(\dfrac{6-5\pi}{8}x\right)

2\cos\left(\dfrac{6+5\pi}{8}x\right) + 2\sin\left(\dfrac{6-5\pi}{8}x\right)

\dfrac{1}{2}\sin\left(\dfrac{6+5\pi}{8}x\right) - \dfrac{1}{2}\sin\left(\dfrac{6-5\pi}{8}x\right)

\cos\left(2x\right)\sin\left(4x\right)\cos\left(4x\right)

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\dfrac{1}{4}\sin\left(10x\right)+\dfrac{1}{4}\sin\left(6x\right)

\dfrac{1}{2}\sin\left(10x\right)+\dfrac{1}{2}\sin\left(6x\right)

\dfrac{1}{4}\cos\left(10x\right)-\dfrac{1}{4}\cos\left(6x\right)

\dfrac{1}{2}\cos\left(10x\right)-\dfrac{1}{2}\cos\left(6x\right)

\sin\left(2x\right)\sin\left(4x\right)

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\dfrac{1}{2}\cos\left(2x\right) - \dfrac{1}{2}\cos\left(6x\right)

\dfrac{1}{2}\cos\left(6x\right) - \dfrac{1}{2}\cos\left(2x\right)

\cos\left(2x\right) - \cos\left(6x\right)

\cos\left(6x\right) -\cos\left(2x\right)

-\sin\left(3x\right)\sin\left(x\right)

\dfrac{1}{2}\cos\left(4x\right) - \dfrac{1}{2}\cos\left(2x\right)

\dfrac{1}{2}\cos\left(2x\right) - \dfrac{1}{2}\cos\left(4x\right)

\cos\left(4x\right) - \cos\left(2x\right)

\cos\left(2x\right) - \cos\left(4x\right)

\cos\left(\dfrac{x}{2}\right)\cos\left(\dfrac{3x}{2}\right)

\dfrac{1}{2}\cos\left(2x\right) + \dfrac{1}{2}\cos\left(x\right)

\dfrac{1}{2}\cos\left(x\right) - \dfrac{1}{2}\cos\left(2x\right)

\dfrac{1}{2}\sin\left(2x\right) - \dfrac{1}{2}\sin\left(x\right)

\dfrac{1}{2}\sin\left(2x\right) - \dfrac{1}{2}\sin\left(x\right)

\sin\left(\dfrac{x}{2}\right)\cos\left(\dfrac{x}{3}\right)

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\dfrac{1}{2}\sin\left(\dfrac{5x}{6}\right) + \dfrac{1}{2}\sin\left(\dfrac{x}{6}\right)

\dfrac{1}{2}\cos\left(\dfrac{5x}{6}\right) + \dfrac{1}{2}\cos\left(\dfrac{x}{6}\right)

\dfrac{1}{2}\sin\left(\dfrac{5x}{6}\right) - \dfrac{1}{2}\sin\left(\dfrac{x}{6}\right)

\dfrac{1}{2}\cos\left(\dfrac{5x}{6}\right) - \dfrac{1}{2}\cos\left(\dfrac{x}{6}\right)

\dfrac{\sqrt{3}}{2}\cos\left(\dfrac{x}{3}\right)

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\dfrac{1}{2}\sin\left(\dfrac{\pi+x}{3}\right) +\dfrac{1}{2}\sin\left(\dfrac{\pi-x}{3}\right)

\dfrac{1}{2}\cos\left(\dfrac{\pi+x}{3}\right) +\dfrac{1}{2}\cos\left(\dfrac{\pi-x}{3}\right)

\dfrac{1}{2}\sin\left(\dfrac{\pi+x}{3}\right) -\dfrac{1}{2}\sin\left(\dfrac{\pi-x}{3}\right)

\dfrac{1}{2}\cos\left(\dfrac{\pi+x}{3}\right) -\dfrac{1}{2}\cos\left(\dfrac{\pi-x}{3}\right)