# How to substitute trigonometry intuitively?

Given a sequence $\{a_i\}$ such that $a_1=1$ and $a_{n+1}=a_n+\sqrt{1+a_n^2} \;\;\;\;\; \forall{n \in {\mathbb{N}}}$

Calculate $$\lim_{n\to \infty}\frac{a_n}{2^n}$$

I had the answer with me. And it was $\frac{2}{\pi}$

So it came obvious to me that I had to use a trigonometric substitution. So I tried $\tan{\theta}$ and it became messy. The sequence was easy to solve on substituting $\cot{\theta}$.

I used $$\cot{\theta}+\csc{\theta}=\cot{\frac{\theta}{2}}$$

where $$\theta=\frac{\pi}{4}$$

Giving $$a_n=\cot{\frac{\frac{\pi}{4}}{2^{n-1}}}$$

and it was easy to solve ahead.

The problem is I am preparing for a competitive exam and I won't have the answer key there. It wasn't at all intuitive to me that I had to go for a trigonometric substitution? Can I approach this question in any other manner?

Also, How do I get an idea where trigonometric substitutions will make the calculations easier?

Let me know, If I am missing some details. I will add them.

Thank you.

And for such a short time I do not know any alternative way to do this. Personally speaking I might let $b_n = a_n 2^{-n}$ and let $n \to \infty$ but this would yield nothing since we would obtain an equality.
As I experienced, $\sqrt {1\pm t^2}$ would make me try trig substitution. Same thing when I encountered some equation taking a similar form to, say formula of double angles and triple angles. Anyway the more you prepared, the more thoughts come up when dealing the test.
In general, it's wise that whenever you see things of the form $\sqrt{x^2\pm a^2}$ or $\sqrt{a^2 - x^2}$ to attempt trigonometric substitutions.