i used a for alpha.
start with cos^2(a) + cos(a) = sin^2(a)
since sin^2(a) = 1 - cos^2(a), you get:
cos^2(a) + cos(a) = 1 - cos^2(a)
subtract 1 from both sides of the equation and add cos^2(a) to both sides of the equation to get:
cos^2(a) + cos^2(a) + cos(a) - 1 = 0
combine like terms to get:
2 * cos^2(a) + cos(a) - 1 = 0
factor this quadratic equation to get:
(2 * cos(a) - 1) * (cos(a) + 1) = 0
solve for cos(a) to get:
cos(a) = .5 or cos(a) = -1
when cos(a) = .5, a = 60 degrees.
that's in the first quadrant.
cosine is positive in the first and fourth quadrant.
equivalent angle in the fourth quadrant is 360 - 60 = 300 degrees.
when cos(a) = .5, a = 60 degrees or 300 degrees.
when cos(a) = -1, a = 180 degrees.
cosine is negative in the second and third quadrants.
equivalent angle in the first quadrant is 180 minus 180 = 0
equivalent angle in the third quadrant is 180 + 0 = 180.
looks like only one angle where cos(a) = -1 between 0 and 360 degrees and that's 180 degrees.
your solution is alpha = 60, 180, or 300 degrees.
when alpha = 60 or 300 degrees, cos(alpha) = .75
when alpha = 180 degrees, cos(alpha) = -1.
note that cos^2(60) + cos(60) = .75 and sin^2(60) also = .75.
note that cos^2(300) + cos(300) = .75 and sin^2(300) also = .75
note that cos^2(180) + cos(180) = 0 and sin^2(180) also = 0
this is seen on the graph.
