If = 12, Prove that (3 -4 ^3 )=0. - Vidyadip

Question

If $\sin\alpha=\frac12,$ Prove that $\big(3\cos\alpha-4\cos^3\alpha\big)=0.$

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Answer

$\sin\alpha=\frac12\Rightarrow\sin^2\alpha=\frac14$
$\therefore\cos^2\alpha=1-\sin^2\alpha=1-\frac14=\frac34$
$\Rightarrow\cos\alpha=\frac{\sqrt{3}}{2}$
$\therefore\ \text{L.H.S.}=3\cos\alpha-4\cos^2\alpha$
$=3\times\frac{\sqrt{3}}{2}-4\Big(\frac{\sqrt{3}}{2}\Big)^2$
$=\frac{3\sqrt{3}}{2}-4\times\frac{3\sqrt{3}}{8}$
$=\frac{3\sqrt{3}}{2}-\frac{3\sqrt{3}}{2}$
$=0$
$=\text{R.H.S}$

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