Make the trigonometric substitution

Question

x = a tan 𝜃 for −𝜋/2 < 𝜃 < 𝜋/2 and a > 0. Simplify the resulting expression. x^2/ sqroot a^2+x^2

Answer 1

Plug in a²tan²(θ) wherever you see x²

Factor out a² under the radical and then take the square root of it to get "a" on the outside of the radical and inside the radical you will have 1 + tan²(x)

Use the Pythagorean Identity sec²(x) = 1 + tan²(x) then take the square root of sec²(x) to eliminate the radical.

Simplify by converting tan²(θ) to tan(θ)•tan(θ) which is tan(θ)•sin(θ)/cos(θ) and by converting 1/sec(θ) to cos(θ). After cancelling several items it simplifies right down.