Under the substitution $x=2$tanθ with θ acute, the expression √$(4+x^2)$ becomes:

√$(4+x^2)$ = √(4 + 4ta$n^2$θ) = 2√(1 + ta$n^2$θ) = 2√(se$c^2$θ) = 2secθ. This is why $x=a$·tanθ is the standard substitution for √$(a^2+x^2)$, while $x=a$·sinθ suits √$(a^2-x^2)$.