Need help understanding $\frac {dx}{\cos^2(\frac{x}{2})} = 2d(\operatorname{tg}(\frac{x}{2})) $

Question

I have found this statement somewhere, however, I dont really understand it.

Could someone explain me where does $2$ before $\operatorname{tg}(x/2)$ come from?

$$\frac {dx}{\cos^2(\frac{x}{2})} = 2d(\operatorname{tg}(\frac{x}{2})) $$

Answer 1

Hint: (Edited to make the spoiler spoil less things!)

• $d(\tan x)=\sec^2 x \rm dx $
• Chain Rule
• $\operatorname{tg}(x)$ is an archaic name for $\tan x$

$d(\tan \dfrac x 2)=\dfrac 1 2\sec^2\dfrac x 2 \mathrm{d}x \implies \dfrac{\mathrm{d}x}{\cos^2 \dfrac{x}{2}}=2\mathrm{d}(\tan \dfrac{x}{2})$