Sums of Cosines into Products with Sum-to-Product Formulas

Text

Write the following quantities as a product of sines and/or cosines:

(a) $\cos\left(2x+\dfrac\pi4\right)+\cos\left(\dfrac\pi6-x\right)$

(b) $\sin\left(2x+\dfrac\pi4\right)+\sin\left(\dfrac\pi6-x\right)$

(c) $\sin\left(2x+\dfrac\pi4\right)-\sin\left(\dfrac\pi6-x\right)$

Solution

Let $p=2x+\dfrac\pi4$ and $q=\dfrac\pi6-x$. Then: $\frac{p+q}{2} = \frac x2+\frac{5\pi}{24}, \qquad \frac{p-q}{2} = \frac{3x}2+\frac\pi{24}.$

(a) Using $\cos p+\cos q = 2\cos\tfrac{p+q}2\cos\tfrac{p-q}2$: $\cos p+\cos q = 2\cos\!\left(\frac x2+\frac{5\pi}{24}\right)\cos\!\left(\frac{3x}2+\frac\pi{24}\right).$

(b) Using $\sin p+\sin q = 2\sin\tfrac{p+q}2\cos\tfrac{p-q}2$: $\sin p+\sin q = 2\sin\!\left(\frac x2+\frac{5\pi}{24}\right)\cos\!\left(\frac{3x}2+\frac\pi{24}\right).$

(c) Using $\sin p-\sin q = 2\cos\tfrac{p+q}2\sin\tfrac{p-q}2$: $\sin p-\sin q = 2\cos\!\left(\frac x2+\frac{5\pi}{24}\right)\sin\!\left(\frac{3x}2+\frac\pi{24}\right).$

$\boxed{\text{the three products above}}$