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Chebyshev polynomials are a sequence of orthogonal polynomials that provide recurrence relations useful for solving polynomials and approximating functions without extensive calculation.

Let \(T_n\) be the nth Chebyshev Polynomial.

If \(T_{90}(x)=\frac{\sqrt3}{2}\), then find the value of \(T_{360}(x)\).

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\[\sum_{k \in S} \cos\left(\frac{2k\pi}{11}\right)\]

Find the value of the summation above where \(S = \{ 1,2,3,5,7,11 \} \): all the prime numbers between 2 and 11, together with 1.

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\[ \large \displaystyle \prod_{n=1}^{2016} \sec\left(\dfrac{n\pi}{2017}\right)\]

Evaluate the product above.

Given that the above is equal to \(a^b\) where a is a prime number, find \(a+b\).

Inspired by this problem when I solved it in 3 steps... All of which were wrong, but somehow I got the right answer.

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\[ \small \left(1 + 2\cos\frac{2\pi}n\right) \left(1 + 2\cos\frac{4\pi}n\right) \left(1 + 2\cos\frac{6\pi}n\right)\cdots\left(1 + 2\cos\frac{2n\pi}n\right)\]

If \(n\) is a prime number larger than 3, then find the value of the expression above.

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Evaluate

\(\displaystyle \sum_{k=1}^{45} \csc^2(2k-1)\)

The angle is given in degrees.

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