other way of finding sin of 18 degrees

It turns out that you can find the value of sin(18°) by the following method:

Notice that by the sin addition formula: Also: Moreover, by the double cosine formula: Now use the Pythagorean identity like this: Now equate the right-hand side of equation 2° with the right-hand side of statement 1°, and substitute for cos(36°): Notice that for t = 1 the equation is satisfied, so: Now notice that 1 > t > 0, because sin(x) is an increasing function when 90 > x > 0 and sin(90°) = 1 > sin(18°) > sin(0°) = 0, so t = sin(18°) = ^{(-1 + √5)}⁄₄ , since that’s the only solutions that meets the criteria

While this method works, it’s not easy to come up with the first equation. I stumbled on it by pure coincidence. I personally prefer the geometrical way of finding the value of sin(18), because it strikes me as the more obvious way of doing it.