Chebyshev Polynomials

Chebyshev Polynomials: Level 4 Challenges


Let TnT_n be the nth Chebyshev Polynomial.

If T90(x)=32T_{90}(x)=\frac{\sqrt3}{2}, then find the value of T360(x)T_{360}(x).

kScos(2kπ11)\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}S = \{ 1,2,3,5,7,11 \} : all the prime numbers between 2 and 11, together with 1.

n=12016sec(nπ2017) \large \displaystyle \prod_{n=1}^{2016} \sec\left(\dfrac{n\pi}{2017}\right)

Evaluate the product above.

Given that the above is equal to aba^b where a is a prime number, find a+ba+b.

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

k=1n(1+2cos2kπn) \prod_{k=1}^{n} \left(1 + 2\cos\frac{2k\pi}n\right)

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


k=145csc2(2k1)\displaystyle \sum_{k=1}^{45} \csc^2(2k-1)

The angle is given in degrees.


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