Proof of $\arctan(x) = \arcsin(x/\sqrt{1+x^2})$

Consider the right angled triangle with sides $1,x,\sqrt{1+x^2}$

Let $\phi$ be the angle opposite to the side of length $x$.

We find that: $$\phi=\arcsin(x/\sqrt{1+x^2})$$ $$\phi=\arctan(x/1)$$

Thus: $$\arcsin(x/\sqrt{1+x^2})=\arctan(x)$$

Calculate the derivative of both sides:

$$(\arctan x)'=\frac{1}{1+x^2}$$

$$\left(\arcsin\frac{x}{\sqrt{1+x^2}}\right)'=\frac{\sqrt{1+x^2}-\frac{x^2}{\sqrt{1+x^2}}}{1+x^2}\cdot\frac{1}{\sqrt{1-\frac{x^2}{1+x^2}}}=$$

$$=\frac{1}{(1+x^2)\sqrt{1+x^2}}\cdot\frac{\sqrt{1+x^2}}{\sqrt 1}=\frac{1}{1+x^2}$$

Since both derivatives are equal the functions are the same up to the sum of a constant:

$$\arctan x=\arcsin\frac{x}{\sqrt{1-x^2}}+C\,\,\,,\,\,C=\,\text{a constant}$$

Finally, to find what $\,C\,$ is you can, for example, input $\,x=0\,$ in the above...

As soon as you see $\arctan x$, draw a right triangle in which the "opposite" side has length $x$ and the "adjacent" side has length $1$. Then the angle to which those are "opposite" and "adjacent" is $\arctan x$.

The Pythagorean theorem then tells you the length of the hypotenuse.

That gives you the sine of the angle, since $\sin=\dfrac{\mathrm{opp}}{\mathrm{hyp}}$.

That tells you what the angle in question is the arcsine of.