Show there exists an infinite number of angles between 0 and p/2 whose sine and cosine are both rational.

Solution

      The slope of the slanting line is t and it is easy to see that it intersects the circle at our mystery point. That is, (x, y) =

        The statement is the same thing as saying that there are an infinite number of points with rational coordinates on the unit circle.
       Let t be a rational number.

       Writing t = u/v, this is essentially the ancient Greek formula for Pythagorean triangles (a, b, c) = (u² - v², 2uv, u² + v²).

       So the point with those coordinates is on the unit circle and has rational coordinates. There are an infinite number of rational numbers, t, so there are an infinite number of points on the unit circle with rational coordinates.
        You might ask where did  the chosen coordinates

Solution

6^x + 6^-x = 12
6^2x + 1 = 12(6^x)
6^2x -12(6^x) + 1 = 0  (let u = 6^x)
u²- 12u = 1 = 0

Following is an interesting geometrical interpretation courtesy of Dick Koch.