Question

Write the following function in the simplest form:

`tan^(-1)  x/(sqrt(a^2 - x^2)), |x| < a`

Answer 1

`tan^(-1)  x/(sqrt(a^2 - x^2))`

Put x = a sin θ

⇒ `x/a = sin θ`

⇒ `θ = sin^(-1) (x/a)`

∴ `tan^(-1)  x/sqrt(a^2 - x^2) `

= `tan^(-1) ((a sin θ)/(sqrt(a^2 - a^2 sin^2 θ)))`

= `tan^(-1) ((asin θ)/(asqrt(1-sin^2 θ))) `

= `tan^(-1) ((asin θ)/(acos θ))`

= `tan^(-1) (tan θ)`

= θ

= `sin^(-1)  x/a`