<s>[INST] How to relate $2\sin(3\pi/8)-2\sin(7\pi/8)$ and $\csc(3\pi/8)$? Trying to simplify $2\sin(3\pi/8)-2\sin(7\pi/8)$ down to $\csc(3\pi/8)$. The two expressions have equal decimal approximations but I'm literally at my wit's end trying to relate them based on trigonometric identities.
 [/INST] $$\sin(7 \pi/8) = \sin(\pi/2 + 3 \pi/8) = \cos(3 \pi/8)$$
\begin{align}
\dfrac{\sin(3 \pi/8) - \sin(7 \pi/8)}{\csc(3 \pi/8)} & = \sin(3 \pi/8) (\sin(3 \pi/8) - \sin(7 \pi/8))\\
& = \sin(3 \pi/8) (\sin(3 \pi/8) - \cos(3 \pi/8))\\
& = \sin^2(3 \pi/8) - \sin(3 \pi/8) \cos(3 \pi/8)\\
& = \underbrace{\dfrac{1 - \cos(3 \pi/4)}{2}}_{\sin^2(\theta) = \frac{1 - \cos(2 \theta)}2} - \underbrace{\dfrac{\sin(3 \pi/4)}2}_{\sin(\theta) \cos(\theta) = \frac{\sin(2 \theta)}2}\\
& = \dfrac{1 + \dfrac1{\sqrt{2}} - \dfrac1{\sqrt{2}}}2\\
& = \dfrac12
\end{align}
which is what we want.
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