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Question
Write the following in the simplest form:
`tan^-1(x/(a+sqrt(a^2-x^2))),-a<x<a`
Solution
Let `x = asintheta`
Now,
`tan^-1{x/(a+sqrt(a^2-x^2))}=tan^-1{(asintheta)/(a+sqrt(a^2-a^2cos^2theta))}`
`=tan^-1{(asintheta)/(a+asqrt(cos^2theta))}`
`=tan^-1{sintheta/(1+costheta)}`
`=tan^-1{(2sin(theta/2)cos(theta/2))/(2cos^2 theta/2)}`
`=tan^-1{tan theta/2}`
`=theta/2`
`=1/2sin^-1(x/a)`
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