The value of 36 ? ( 4 cos 2 9 ° - 1 ) ( 4 cos 2 27 ° - 1 ) ( 4 cos 2 81 ° - 1 ) ( 4 cos 2 243 ° - 1 ) is [2023]

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Solution

Q.
The value of $36 (4 \cos^{2} 9 ° - 1) (4 \cos^{2} 27 ° - 1) (4 \cos^{2} 81 ° - 1) (4 \cos^{2} 243 ° - 1)$ is [2023]

1 18

2 54

3 27

4 36

Ans.
(4)

We have, $4 \cos^{2} θ - 1 = 4 (1 - \sin^{2} θ) - 1 = 3 - 4 \sin^{2} θ = \frac{\sin 3 θ}{\sin θ}$

$So, given expression can be written as,$

$= 36 \times \frac{\sin 27 °}{\sin 9 °} \times \frac{\sin 81 °}{\sin 27 °} \times \frac{\sin 243 °}{\sin 81 °} \times \frac{\sin 729 °}{\sin 243 °} = 36 \times \frac{\sin 729 °}{\sin 9 °} = 36$

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