Product into a Sum with Associated Angles and Werner

Text

Write the following quantity as a sum of sines and/or cosines: $\cos\left(\dfrac\pi2-x\right)\sin\left(2x+\dfrac\pi4\right).$

Solution

Using the associated-angles formula $\cos\left(\dfrac\pi2-x\right)=\sin x$, the quantity becomes: $\sin x\,\sin\!\left(2x+\frac\pi4\right).$ Apply the Werner formula $\sin A\sin B = \tfrac12\left[\cos(A-B)-\cos(A+B)\right]$ with $A=x$ and $B=2x+\dfrac\pi4$: $\sin x\,\sin\!\left(2x+\frac\pi4\right) = \frac12\left[\cos\!\left(x+\frac\pi4\right)-\cos\!\left(3x+\frac\pi4\right)\right].$

$\boxed{\tfrac12\left[\cos\!\left(x+\tfrac\pi4\right)-\cos\!\left(3x+\tfrac\pi4\right)\right]}$