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Simplify y=tan^(-1)(x/(1+sqrt(1-x^2)))...

Simplify `y=tan^(-1)(x/(1+sqrt(1-x^2)))`

Text Solution

Verified by Experts

The correct Answer is:
`(1)/(2sqrt(1-x^(2)))`
`y=tan^(-1)""((x)/(1+sqrt(1-x^(2))))`
Put `x= sin theta.` Then,
`y=tan^(-1)((sintheta)/(1+sqrt(1-sin^(2)theta)))=tan^(-1)((sintheta)/(1+cos theta))`
`=tan^(-1)""(2sin""(theta)/(2)cos""(theta)/(2))/(2cos^(2)""(theta)/(2))=tan^(-1)tan""(theta)/(2)=(theta)/(2)`
`"So, "y=(sin^(-1)x)/(2)or(dy)/(dx)=(1)/(2sqrt(1-x^(2)))`
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