Cryptic Cotangent Products
X = k = 1 ∏ 3 0 1 5 cot [ 3 π ( 1 − 3 3 0 1 5 − 1 3 k ) ]
and
Y = k = 1 ∏ 3 0 1 5 cot [ 3 π ( 2 1 − 3 3 0 1 5 − 1 3 k ) ]
Evaluate ( Y 2 X ) 4
The answer is 16.
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It was damn hard to solve. As I am super lazy I can't write whole solution but can give you the recipe and in turn you can find the answer. Please comment if anything is wrong. First step is to evaluate X / Y and when we try to do that convert the cot function in the numerator to tan by general conversion c o t ( x ) = t a n ( 9 0 − x ) But we'll use pi/2 instead of 90. And substitute the exponent to theta. Also convert the cot function to tan by reciprocating it. And in the end we get something like t a n ( 3 0 + x ) t a n ( 3 0 − x ) which we can evaluate and then we get a telescoping product. Just to make sure that you are correct see that there must be a term like 3 − t a n 2 ( x ) in denominator if you are following my method. And then use the formula of t a n ( 3 x ) to substitute in the numerator and we see a magic cancellation .