# Solve system of equations with sin, cos, tg

I am trying to solve this system of equations but without any results.

How can I solve this system of equations (in real numbers)?

$$\sin^2 x + \cos^2 y = \tan^2 z$$

$$\sin^2 y + \cos^2 z = \tan^2 x$$

$$\sin^2 z + \cos^2 x = \tan^2 y$$

The inherent symmetry in these equations suggests trying whether there are solutions of the form $x=y=z$. Of course, there might be other solutions as well. – Christian Blatter Dec 2 '12 at 15:14
Try converting each term to sine: $$\sin^2x + (1-\sin^2y) = {\sin^2z \over 1-\sin^2z}$$ $$\sin^2y + (1-\sin^2z) = {\sin^2x \over 1-\sin^2x}$$ $$\sin^2z + (1-\sin^2x) = {\sin^2y \over 1-\sin^2y}$$ If you substitute A, B, and C for $\sin^2x, \sin^2y, \sin^2z$, you'll have three equations with three unknowns, so you should be able to solve from there.
Note that if you just write out function names like $\sin$, they get interpreted as juxtaposed variable names and formatted (e.g. italicized) accordingly. To get the proper font and spacing for such functions, you can use the predefined commands like \sin, or if you need a function for which there's no predefined command, you can use \operatorname{name}. – joriki Dec 2 '12 at 15:47