Impossible identity? $ \tan{\frac{x}{2}}$ $$\text{Let}\;\;t = \tan\left(\frac{x}{2}\right). \;\;\text{Show that}\;\dfrac{dx}{dt} = \dfrac{2}{1 + t^2}$$
I am saying that this is false because that identity is equal to $2\sec^2 x$ and that can't be equal. Also if I take the derivative of an integral I get the function so if I take the integral of a derivative I get the function also so the integral of that is $x + \sin x$ which evaluated at 0 is not equal to $\tan(x/2)$

You're wrong, the identity is correct.  Note that $t = \tan(x/2)$ implies
$x = 2 \arctan(t) + 2 n \pi$.
