Let = (1 - 2 ( 11 ) ) (1 - 2 ( 3 11 ) ) (1 - 2 ( 9 11 ) ) (1 - 2 ( 27 11 ) ) (1 - 2 ( 81 11 ) ) . Then the value of 5 - ^2 is _____________.

Step-by-step solution

Let the given expression be . First, we simplify the angles in the product. ( 27 11 ) = (2 + 5 11 ) = ( 5 11 ) ( 81 11 ) = (7 + 4 11 ) = - ( 4 11 ) = ( - 4 11 ) = ( 7 11 ) Thus, the angles are 11 , 3 11 , 5 11 , 7 11 , 9 11 , which can be written as _k = (2k-1) 11 for k = 1, 2, 3, 4, 5 . The expression becomes: = _ k=1 ⁵ (1 - 2 _k ) Using the trigonometric identity 1 - 2 = 1 - 2(2 ^2( /2) - 1) = 3 - 4 ^2( /2) . Multiplying and dividing by ( /2) , we get: 1 - 2 = 3 ( /2) - 4 ^3( /2) ( /2) = - (3 /2) ( /2) Applying t