How to find the solutions to sin^2-cos^2=1-cos

You can use a graphing calculator, or you can replace sin^2 by 1-cos^2 and solve the quadratic.
.. (1-cos^2) -cos^2 = 1 -cos
.. -2cos^2 = -cos
.. cos(2cos -1) = 0
The zero-product rule tells you
.. cos = 0 . . . . . . angle = π/2 +kπ
.. cos = 1/2 . . . . . angle = ±π/3 +2kπ