is this legal?

Yup
cos²(u) + sin²(u) = 1
cos²(2t) + sin²(2t) = 1
cos²(pi/17) + sin²(pi/17) = 1
cos²(3x²+6x+1) + sin²(3x²+6x+1) = 1

And...

 I blew up some other alternate dimension, not this one... I swear

cos²(1/0) + sin²(1/0) = 1?! .......osht

1/0 is not a definite value, thus it cannot be given the state of equality with another 1/0 inside the arguments of those trigonometric functions, and the principle cannot be applied.
For the principle to work, 1/0 must equal 1/0, which is not true.
1/0 != 1/0

Does 0/0 = 1? =D

what about infinity/infinity? (can't do sideways "8" D=)

0/0 is undefined. Reason being, if we set some number equal to 0/0:
x = 0/0
Suppose we could multiply both sides by 0, getting:
0x = 0
x = lR?
But even that's bad math because you can't just eliminate the 0's in 0/0 * 0 and call it 0.

infinity is undefined, so infinity/infinity is too...

relax man. im not quite here yet.

 Spoiler

If I am NOT mistaken I do believe that is lake mead >_> I could be wrong though o.o