Sum of Cosines into a Product (Sum-to-Product)

Text

Write the following quantity as a product of sines and/or cosines: $\cos\left(\dfrac x2+\dfrac\pi3\right)+\cos\left(\dfrac\pi4-2x\right).$

Solution

Let $p=\dfrac x2+\dfrac\pi3$ and $q=\dfrac\pi4-2x$. Then: $\frac{p+q}{2} = -\frac{3x}4+\frac{7\pi}{24}, \qquad \frac{p-q}{2} = \frac{5x}4+\frac\pi{24}.$ Using $\cos p+\cos q = 2\cos\tfrac{p+q}2\cos\tfrac{p-q}2$ together with the evenness of cosine, $\cos(-\theta)=\cos\theta$: $\cos p+\cos q = 2\cos\!\left(\frac{3x}4-\frac{7\pi}{24}\right)\cos\!\left(\frac{5x}4+\frac\pi{24}\right).$

$\boxed{2\cos\!\left(\tfrac{3x}4-\tfrac{7\pi}{24}\right)\cos\!\left(\tfrac{5x}4+\tfrac\pi{24}\right)}$