Putnam Math Exam Problem: Evaluate [MATH]\prod_{n=2}^{\infty}\frac{n^3-1}{n^3+1}[/MATH]

Putnam Math Exam Problem:

Evaluate [MATH]\prod_{n=2}^{\infty}\frac{n^3-1}{n^3+1}[/MATH].

I admit it, the first (or should I say, the only) thing that we could do about the given fraction's expression, is to factor it:

Simplify $\dfrac{\cos 1^{\circ}+\cos 2^{\circ}+\cdots+\cos 44^{\circ}}{\sin 1^{\circ}+\sin 2^{\circ}+\cdots+\sin 44^{\circ}}$

Express $\dfrac{\cos 1^{\circ}+\cos 2^{\circ}+\cdots+\cos 44^{\circ}}{\sin 1^{\circ}+\sin 2^{\circ}+\cdots+\sin 44^{\circ}}$ in the form $a+b\sqrt{c}$, where $a,\,b,\,c$ are positive integers.

(This question showed up here(math-teacher-guide) but I did not include its solution in that slide show).

My solution:

Normally for simplifying trigonometric question such as this one, there is one thing that's so worth noticing:

IMO Practice Problem: Find the exact real root for the equation $10x^3-12x^2-6x-1=0$

Find the exact real root for the equation $10x^3-12x^2-6x-1=0$.

The given cubic cannot be factored easily and beautifully, so the method of factoring the given polynomial is out of the question.

I hear you, the next best approach might be to try out the substitution method, with the hope that after the substitution, we have less variable terms in our newly set equation. But there seems no suitable substitution is available so to make simpler the given equation.

Olympiad Trigonometry Problem

Prove $\tan^2 20^\circ+\tan^2 40^\circ+\tan^2 80^\circ=33$.

Good trigonometry problem is hard to come by, and when we, the math educator found one, we have to take advantage of it and make full use of it.

Of course, we can rest assured that this trigonometry problem can be tackled using the sum-to-product and product-to-sum identities in a really messy and tedious way. After all, any given math problem can be solved in the most traditional way, isn't it?