<s>[INST] Find $d$ when $(\sin x)^7 = a \sin 7x + b \sin 5x + c \sin 3x + d \sin x$ There exist constants $a$, $b$, $c$, and $d$ such that
$(\sin x)^7 = a \sin 7x + b \sin 5x + c \sin 3x + d \sin x$
for all angles $x$. Find $d$.
 [/INST] If you only need to know what $d$ is, then consider $z=\cos\theta+i\sin\theta$, in which case $$\left(z-\frac 1z\right)^7=(2i\sin\theta)^7=...+35z^4\left(-\frac 1z\right)^3+35z^3\left(-\frac 1z\right)^4+...$$
Hence
$$2^7i^7\sin^7\theta=...-35\left(z- \frac 1z\right)...=...-35.2i\sin\theta+...$$
therefore the required $$d=\frac{35}{64}$$
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