<s>[INST] Express $\frac{\sin 7\theta}{\sin \theta}$ in powers of $\sin \theta$ only By using DeMoivre's theorm express
$$\frac{\sin 7\theta}{\sin \theta}$$
in the powers of Sine only
answer given in the book is
$$7-56\sin ^2\theta+112\sin ^4 \theta-64\sin^6 \theta$$
can any one help to solve the question
 [/INST] $(\cos x+i\sin x)^7=\cos 7x+i\sin7x$.
We have
$$(r+t)^7=r^7+7r^6t+21r^5t^2+35r^4t^3+35r^3t^4+21r^2t^5+7rt^6+t^7$$
 hence, making $\sin x=t$ and $\cos x=r$ and having in account the powers of $i$, we get
 $$\sin 7x=7r^6t-35r^4t^3+21r^2t^5-t^7$$ so, because of $\cos^2x=1-\sin^2x$,
$$7t-21t^3+21t^5-7t^7-35t^3+70t^5-35t^7+21t^5-21t^7-t^7=-64t^7+112t^5-56t^3+7t$$
Now dividing by $t$, we finish.
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