<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] Evaluate the equation for different $x$:
$$\begin{align}\\
\frac\pi6&\to -a+b+2c&+d&=\frac1{64}\\
\frac\pi4&\to -a-b+\ c&+d&=\frac8{64}\\
\frac\pi3&\to +a-b&+d&=\frac{27}{64}\\
\frac\pi2&\to -a+b-\ c&+d&=\frac{64}{64}\\
\end{align}$$
(after normalization of the coefficient of $d$).
Elimination by the combination $(1)+3\times(3)+2\times(4)$ yields
$$6d=\frac{210}{64}.$$
The evaluation for $x=\dfrac\pi4$ was not even necessary.
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