The value of 36(4 ^29^ -1)(4 ^227^ -1)(4 ^281^ -1)(4 ^2243^ -1) is :

Question

The value of $36(4{\cos }^{2}{9}^{\circ }-1)(4{\cos }^{2}{27}^{\circ }-1)(4{\cos }^{2}{81}^{\circ }-1)(4{\cos }^{2}{243}^{\circ }-1)$ is :

Accepted Answer

$$
\begin{aligned}
& 4 \cos ^2 \theta-1=4\left(1-\sin ^2 \theta\right)-1 \\
& =3-4 \sin ^2 \theta \\
& =\frac{3 \sin \theta-4 \sin ^3 \theta}{\sin \theta} \\
& =\frac{\sin 3 \theta}{\sin \theta}
\end{aligned}
$$

$$36\left(4 \cos ^{2} 9^{\circ}-1\right)\left(4 \cos ^{2} 27^{\circ}-1\right)\left(4 \cos ^{2} 81^{\circ}-1\right)\left(4 \cos ^{2} 243^{\circ}-1\right)$$

$$
\begin{aligned}
& =36\left[\frac{\sin 27^{\circ}}{\sin 9^{\circ}} \times \frac{\sin 81^{\circ}}{\sin 27^{\circ}} \times \frac{\sin 243^{\circ}}{\sin 81^{\circ}} \times \frac{\sin 729^{\circ}}{\sin 243^{\circ}}\right] \\
& =36\left[\frac{\sin 729^{\circ}}{\sin 9^{\circ}}\right]=36 \times 1=36
\end{aligned}
$$

Solutions · 1