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Question
Write the following function in the simplest form:
`tan^(-1) x/(sqrt(a^2 - x^2)), |x| < a`
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Solution
`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`
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