Recently while studying Elliptic Integrals and related topics I have come across various Interesting Trigonometric Substitutions, examples given below
1.
$$\int_0^{\pi/2}\frac{1}{\sqrt{1-x^2(\sin t)^4}}\ dt=\int_0^{\pi/2}\frac{1}{\sqrt{1+x-2x(\sin v)^2}}\ dv$$
which uses the substitution
$$v=\arctan(\sqrt{1+x}\tan t)$$
- The well known Landen's Transformation $$\int_{0}^{\pi/2}\frac{1}{\sqrt{1-x^{2}(\sin t)^2}}\ dt=\int_{0}^{\pi/2}\frac{1}{\sqrt{\left(1+x\right)^{2}-4x(\sin v)^2}}\ dv$$ which uses the substitution
$$2v=t+\arcsin(x\sin t)$$
- Landen's Transformation but different proof
$$\int_0^{\pi/2}\frac{1}{\sqrt{(\cos t)^2+x^2(\sin t)^2}}\ dt=\int_{-\pi/2}^{\pi/2}\frac{1}{\sqrt{(1+x)^2-(1-x)^2(\sin v)^2}}\ dv$$
which uses the substitution
$$t=\arctan\left(\frac{\tan v+\sec v}{\sqrt{x}}\right)$$
- Gauss's Substitution for Invariance of Cayley Elliptic Integral under AGM
$$\int_0^{\pi/2}\frac{1}{\sqrt{(a\cos t)^2+(b\sin t)^2}}\ dt=\int_0^{\pi/2}\frac{1}{\sqrt{(\frac{a+b}{2}\cos v)^2+(\sqrt{ab}\sin v)^2}}\ dv$$ which uses the substitution
$$\sin t=\frac{2a\sin v}{a+b+(a-b)(\sin v)^2}$$
These are the only examples I could recall right now but there would be many others.
As I am inexperienced in these types of substitutions, to me all of them feel like they were pulled out of thin air.
Even after the substitutions a good amount of algebraic manipulations are needed.
Is there a standard way I can learn and understand all of these substitutions and why they work?
