The value of 364cos2⁡9∘−14cos2⁡27∘−14cos2⁡81∘−14cos2⁡243∘−1 is

364cos2⁡9∘−14cos2⁡27∘−14cos2⁡81∘−14cos2⁡243∘−1364cos2⁡9−14cos2⁡27−14cos2⁡81∘−14cos2⁡63∘−13621+cos⁡18∘−121+cos⁡54∘−121+cos⁡162∘−121+cos⁡126∘−1361+2cos⁡18∘1+2cos⁡54∘1-2cos⁡18∘1−2cos⁡54∘=361−4cos2⁡18∘1−4cos2⁡54∘361−4⋅10+25161−4⋅10−251636(6+25)4(6−25)4=36

The value of 364cos2⁡9∘−14cos2⁡27∘−14cos2⁡81∘−14cos2⁡243∘−1 is

364cos2⁡9∘−14cos2⁡27∘−14cos2⁡81∘−14cos2⁡243∘−1364cos2⁡9−14cos2⁡27−14cos2⁡81∘−14cos2⁡63∘−13621+cos⁡18∘−121+cos⁡54∘−121+cos⁡162∘−121+cos⁡126∘−1361+2cos⁡18∘1+2cos⁡54∘1-2cos⁡18∘1−2cos⁡54∘=361−4cos2⁡18∘1−4cos2⁡54∘361−4⋅10+25161−4⋅10−251636(6+25)4(6−25)4=36

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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

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answer is B.

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Detailed Solution

$36 (4 \cos^{2} 9^{\circ} - 1) (4 \cos^{2} 27^{\circ} - 1) (4 \cos^{2} 81^{\circ} - 1) (4 \cos^{2} 243^{\circ} - 1)$
$\begin{array}{l} 36 (4 \cos^{2} 9 - 1) (4 \cos^{2} 27 - 1) (4 \cos^{2} 81^{\circ} - 1) (4 \cos^{2} 63^{\circ} - 1) \\ 36 [2 (1 + \cos 18^{\circ}) - 1] [2 (1 + \cos 54^{\circ}) - 1] [2 (1 + \cos 162^{\circ}) - 1] [2 (1 + \cos 126^{\circ}) - 1] \\ 36 (1 + 2 \cos 18^{\circ}) (1 + 2 \cos 54^{\circ}) (1 - 2 \cos 18^{\circ}) (1 - 2 \cos 54^{\circ}) \\ = 36 (1 - 4 \cos^{2} 18^{\circ}) (1 - 4 \cos^{2} 54^{\circ}) \\ 36 (1 - 4 \cdot \frac{10 + 2 \sqrt{5}}{16}) (1 - 4 \cdot \frac{10 - 2 \sqrt{5}}{16}) \\ 36 \frac{(6 + 2 \sqrt{5})}{4} \frac{(6 - 2 \sqrt{5})}{4} = 36 \end{array}$