I got this when I did it.
[tex] T_1 = \frac{mg - T_2sin\theta_2}{sin\theta_1} [/tex]
Surely [tex] cos(90 - \theta_1) [/tex] is the same as [tex] sin(\theta_1) [/tex]?

The expression you gave in your original post was re-arranged wrong, but the correct one is the same as the one I posted earlier. To get rid of theata2, resolve vertically and substitute in.

in the link that you gave, theta2 was 45. The answer includes a tangent of theta2, but since tangent of 45 is just 1, it does not appear in the answer there.