Let a function f(x) be defined as f(x) = _ k=1 ⁸ cosec ( k 3 + x ) cosec ( (k+1) 3 + x ) . The value of |f( 12 )| is equal to :

Step-by-step solution

The given function is f(x) = _ k=1 ⁸ 1 ( k 3 + x ) ( (k+1) 3 + x ) . The difference between the angles in the denominator is ( (k+1) 3 + x ) - ( k 3 + x ) = 3 . Multiply and divide the expression by ( 3 ) : f(x) = 1 ( 3 ) _ k=1 ⁸ ( ( (k+1) 3 + x ) - ( k 3 + x ) ) ( k 3 + x ) ( (k+1) 3 + x ) Using the identity (A - B) = A B - A B , each term in the sum splits into a difference of cotangents: f(x) = 2 3 _ k=1 ⁸ [ ( k 3 + x ) - ( (k+1) 3 + x ) ] This is a telescoping series, so all intermediate terms cancel out, leavi