How do you calculate cos squared theta equals one minus twelve thirteenths squared?

Answer 1

Writing x instead of theta, cos2x = 1 - (12/13)2 = 1 - 144/169 = 25/169 = (5/13)2

So cos(x) = ± 5/13

so that x = cos-1(5/13) or cos-1(-5/13)

And then, depending on the range of x, you have solutions for x. A calculator will only give you the principal solutions, though.