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Find the derivatives of the following : ...

Find the derivatives of the following :
`cos(2tan^(-1)sqrt((1-x)/(1+x)))`

A

`1-(1)/(sqrt(1-x^(2)))`

B

`1-(1)/(sqrt(1+x^(2)))`

C

`2-(1)/(sqrt(1-x^(2)))`

D

`2-(1)/(sqrt(1+x^(2)))`

Text Solution

Verified by Experts

The correct Answer is:
A
Let `x=cos theta, therefore theta=cos^(-1)x`
`therefore" "y=cos (2 tan^(-1)sqrt((1-x)/(1+x)))-2cos^(-1).sqrt((1-x)/(2))`
`=cos(2tan^(-1)(tan.(theta)/(2)))-2 cso^(-1)(sin.(theta)/(2))`
`=cos theta-2((pi)/(2)-sin^(-1)(sin.(theta)/(2)))`
`=cos theta-pi+theta`
`therefore" "y=x-pi+cos^(-1)x`
`therefore" "(dy)/(dx)=1-(1)/(sqrt(1-x^(2)))`
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