Converst trigonometrical sum in product I'm starting to work with some trigonometrical properties and I got the following problem: Convert the following sum into a product: $\cos{x} + \cos{3x} + \cos{5x} + \cos{7x}$ Using the property with states: $\cos{\alpha} + \cos{\beta} = 2 \times \cos{\frac{\alpha + \beta}{2}} \times \cos{\frac{\alpha - \beta}{2}}$. I got: $2 \times \cos^2{2x} + 2 \times \cos{6x} \times \cos{x}$ How to proceed now ?

Using Prosthaphaeresis Formulas,

$$\cos x+\cos7x=2\cos3x\cos4x$$ and $$\cos3x+\cos5x=2\cos x\cos4x$$

Taking out $2\cos4x,$ $$\cos x+\cos3x=?$$