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# Chebyshev Polynomials

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

# Chebyshev Polynomials: Level 4 Challenges

Let $$T_n$$ be the nth Chebyshev Polynomial.

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

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

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.

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

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.

$\prod_{k=1}^{n} \left(1 + 2\cos\frac{2k\pi}n\right)$

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

Evaluate

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

The angle is given in degrees.

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