The value of 36 (4 ^ 2 9 ^ ∘ - 1) ( 4 ^ 2 27 ^ ∘ - 1)( 4 ^ 2 81 ^ ∘ - 1)(4 ^ 2 243 ^ ∘ - 1) is

Correct answer: D. 36 We need to find the value of 36 (4 ^ 2 9 ^ ∘ - 1) ( 4 ^ 2 27 ^ ∘ - 1)( 4 ^ 2 81 ^ ∘ - 1)(4 ^ 2 243 ^ ∘ - 1) = 36 [ (4 ^ 2 9 ^ ∘ - 1) ( 4 ^ 2 9 ^ ∘ - 1) (4 ^ 2 27 ^ ∘ - 1)(4 ^ 2 27 ^ ∘ - 1)] = 36[ (4 (2cos 9 ^ ∘ 9 °)2 - 4 ^ 2 9 ° - 4 ^ 2 9 ° + 1) (4 (2sin 27 ^ ∘ 9 °)2 - 4 ^ 2 27 ° - 4 ^ 2 27 ° + 1)] = 36 [ ( 4 ^ 2 18 ^ ∘ - 4 + 1)(4 ^ 2 54 ^ ∘ - 4 + 1)] = 36 [ ( 4 ^ 2 18 ^ ∘ - 3)( 4 ^ 2 54 ^ ∘ - 3)] = 36[\ ( 5 -1)2 4 -3\ \ ( 5 + 1)2 4 - 3\ ] = 36 [ \ (6 - 2 5 -12)4(6 + 2 5 -12)4\ ] = - 36[\ (2 5 )2 - 36 16 \ ] =36 Hence this is the correct option.

JEE Main

Exam: JEE Main
Subject: Mathematics
Chapter: Trigonometric Ratios & Identities

JEE Main · Mathematics · Trigonometric Ratios & Identities

The value of 36 (4 ^ 2 9 ^ ∘ - 1) ( 4 ^ 2 27 ^ ∘ - 1)( 4 ^ 2 81 ^ ∘...

The value of

36 \left(4\cos^{2}{9}^{∘} - 1\right) \left(

4\cos^{2}{27}^{∘} - 1\right)\left(

4\cos^{2}{81}^{∘} - 1\right)\left(4\cos^{2}{243}^{∘} - 1\right) is

Options

A $54$
B $18$
C $27$
D $36$

Correct answer

D. $36$

Step-by-step solution

We need to find the value of
$36 \left(4\cos^{2}{9}^{∘} - 1\right) \left($ 4\cos^{2}{27}^{∘} - 1\right)\left( $4\cos^{2}{81}^{∘} - 1\right)\left(4\cos^{2}{243}^{∘} - 1\right)$
$= 36 \left[ \left(4\cos^{2}{9}^{∘} - 1\right) \left($ 4\sin^{2}{9}^{∘} - 1\right) \left(4\cos^{2}{27}^{∘} - 1\right)\left(4\sin^{2}{27}^{∘} - 1\right)\right] $$ = 36\left[ \left(4 \left(2cos{9}^{∘}\sin 9 °\right)2 - 4\cos^{2}9 ° - 4\sin^{2}9 ° + 1\right) \left(4 \left(2sin{27}^{∘}\cos 9 °\right)2 - 4\cos^{2}27 ° - 4\sin^{2}27 ° + 1\right)\right] $$ = 36 \left[ \left( $4\sin^{2}{18}^{∘} - 4 + 1\right)\left(4\sin^{2}{54}^{∘} - 4 + 1\right)\right]$
$= 36 \left[ \left($ 4\sin^{2}{18}^{∘} - 3\right)\left( $4\sin^{2}{54}^{∘} - 3\right)\right]$
$= 36\left[\left\{\dfrac{ \left(\sqrt{5} -1\right)2}{4} -3\right\} \left\{\dfrac{\left(\sqrt{5} + 1\right)2}{4} -$ 3\right\} $\right]$
$= 36 \left[ \left\{ \left(6 - 2\sqrt{5} -12\right)4\left(6 + 2\sqrt{5} -12\right)4\right\}\right]$
$= -$ 36\left[\left\{\dfrac{\left(2\sqrt{5}\right)2 - 36}{16}\right\}\right] =36 $Hence this is the correct option.</blockquote>$

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