S3E3: One telescope, two telescopes

Starting with this episode we step up close and personal with the idea that fixing a problem and constructing as many distinct enough solutions for it as possible should be an integral part of the fabric of studying and doing mathematics (and physics, and computer science).

As such, across the next three episodes our sample "fixed problem" will be the task of compressing a certain finite trigonometric sum into a product.

In the spirit of the above "one problem, many solutions" doctrine, in this episode, see* below, we develop an algebraic solution of the standing problem using nothing more than the middle school trigonometric arithmetic and, along the way, we get acquainted with the so-called "telescopic" sums.

First, we discuss a minimal viable amount of basic theory by looking at a plausible definition of a telescopic sum.

Next, we look at a concrete (popular) kindergarten variety telescopic sum in order to dip our toe into the waters of telescopic sums.

Next, we take the just learned notion for a test drive and find a way to expose the telescopic nature of a, mildly, cleverly modified sum of interest.

Next, we witness step-by-step exactly how the telescopicity of the modified sum comes to life and how that telescopicity manifests itself.

Next, we cancel the modified summands in bulk, take the only two such summands left standing and fold them into a product.

Thus producing the sought-after "closed form formula" that compresses the given finite trigonometric sum into a corresponding product.

We develop our success by exploiting the new result in a number of ways.
Lastly, we look at one practical application of the obtained result (integration in the domain of real numbers).

*In the next, S3E4, or the "A Complex Approach" episode, we will obtain the same results algebraically as well but this time we will do so by using the tactical might of complex numbers.

In the following, S3E5, or the "Euclide Says" episode, we will share with our audience a solution of the current problem that is based solely on synthetic Euclidean plane geometry.