Write the following function in the simplest form: `tan^(-1)(sqrt(1+x^

1.	Write the following function in the simplest form: `tan^(-1)(sqrt(1+x^2)-1)/x , x!=0`
Answer» `tan^(-1)""(sqrt(1+c^(2))-1)/(x) " " Let x=tan theta implies tan^(-1)x=theta` `=tan^(-1)""(sqrt(1+tan^(2))-1)/(tan theta)` `tan^(-1)((sec theta-1)/(tan theta))=tan^(-1)(((1)/(cos theta)-1)/((sin theta)/(cos theta)))` `=tan^(-1)((1-cos theta)/(sin theta))=tan^(-1)((2 sin^(2)""(theta)/(2))/(2 sin ""(theta)/(2) cos ""(theta)/(2)))` `tan^(-1)(tan""(theta)/(2))=(1)/(2) theta=(1)/(2) tan^(-1)x`

Discussion

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