Solve the following equations given that 0 < x < 2pi

For the first one, we have

cos²x - sin²x = -1

Note that sin²x = 1 - cos²x       So we have

cos²x - (1 - cos²x ) = -1

2cos²x - 1 = -1            add 1 to both sides

2cos²x = 0                   divide by 2 on both sides

cos²x = 0                     take the suare root of both sides

cosx = 0                       So x = pi/2 and x =3pi/2

For the second one, we have

1 + √(3) tan(2x)  = 0       Subtract 1 from both sides

√(3) tan(2x)  =-1             Divide both sides by √(3)

tan(2x) = -1/√(3)

Note that tan(x) = -1/√(3) at  150 degrees and at 330 degrees

So tan(2x) = -1/√(3) at 75 degrees and at 165 degrees

And converting these to rads, we have x = (5pi)/12 and x = (11pi)/12

Thanks CPhill!