Beautiful Trigonometry

The purpose of this paper is to illustrate how the Difference Identity for Cosine can be used.

Let us start here:

cos (A - B) = cos (A) cos (B) + sin (A) sin (B)

As a quick corollary notice that,

sin (A + B) = cos [(90° - (A + B)] = cos [(90° - A) - B] = cos (90° - A) cos (B) + sin (90° - A) sin (B) = sin (A) cos (B) + cos (A) sin (B)

where lines (1) and (4) follow since

sin (θ) = cos (90° - θ)

Thus,

sin (A - B) = sin (A) cos (B) - cos (A) sin (B)

We can also add or subtract $\cos(A+B)$ and $\cos(A-B)$ together. This gives,

cos (A) cos (B) = \frac{1}{2} [cos (A + B) + cos (A - B)]

and,

sin (A) sin (B) = \frac{1}{2} [cos (A - B) - cos (A + B)]

As a quick application consider the integral

\int_{0}^{π} sin^{2} (x) d x

With the formula's above we can rewrite our integral
as follows:

\int_{0}^{π} sin^{2} (x) d x = \int_{0}^{π} sin (x) sin (x) d x = \frac{1}{2} \int_{0}^{π} cos (0) d x - \frac{1}{2} \int_{0}^{π} cos (2 x) d x = \frac{1}{2} \int_{0}^{π} d x - \frac{1}{4} \int_{0}^{π} cos (u) d u = \frac{π}{2} - \frac{1}{4} [sin (π) - sin (0)] = \frac{π}{2}