I saw a youtube Video, where someone solves the Gaussian Integral
$$\int_0^\infty e^{-x^2} dx = \frac{\sqrt\pi}2$$
Here is the link: https://www.youtube.com/watch?v=_7TlAzfs_9Q&t=52s (My problem starts at 7:38)
I am not really interested in the Equality of that, but on a Substitution being used in order to solve this. He uses different Equations as 'tools' in order to show that the Equation before is true and uses substitution for that.
This is the Tool-Equation: $$\lim_{ n\to\infty}\sqrt{n}\int_0^\infty e^{(1-x^2)^n} dx$$
Let $x = \sin(\theta)$, which makes $dx = \cos(\theta)d\theta$.
I'm not that well versed on trigonometry so I don't grasp why $dx = \cos(\theta)d\theta$ instead of $dx = d\sin(\theta)$. What are the trigonometric rules to understand that?