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Question
Prove that:
`tan^(-1) sqrtx = 1/2 cos^(-1) ((1-x)/(1+x)) , x in [0, 1]`
Solution
Let x = `tan^2 theta` Then `sqrtx= tan theta`
=> `theta = tan^(-1) sqrtx`
`:. (1-x)/(1+x) `
=` (1-tan^2 theta)/(1+tan^2 theta) `
= `cos 2 theta`
Now we have,
R.H.S = `1/2 cos^(-1) ((1-x)/(1+x)) `
`= 1/2 cos^(-1)(cos 2 theta) `
`= 1/2 xx 2theta `
`= theta = tan^(-1) sqrtx` = L.H.S
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