Hello from MrBillDoesMath!
Answer: (sin(@)) ^2 + (cos(@)) ^2 = 1 for all angles @.
Discussion ;
Take a point, say in the first quadrant. on a unit circle (circle with radius 1) Let that point by (x,y). The from the Pythagorean theorem.
x^2 + y^2 = r^2 or as r = 1
x^2 + y ^2 = 1 ("Equation 1")
If you drop a line from (x,y) perpendicular to the x axis you find that
y = r sin@ or y = sin@ as r =1.
and
x = r cos@ or x = cos@ as r = 1
(Look at the triangle formed by the radius, the perpendicular line and the positive x axis). Substitute these x and y values in Equation 1 above. This gives
(sin@)^2 + (cos@)^2 =1
which is true for all angles. This is a key "pattern" or identity that you should commit to long term memory!
Regards, MrB