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Write the Function in the Simplest Form: `Tan^(-1) (Sqrt(1+X^2) -1)/X, X != 0` - Mathematics

Write the function in the simplest form: `tan^(-1)  (sqrt(1+x^2) -1)/x, x != 0`

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Solution

`tan^(-1)  (sqrt(1+x^2) -1)/x`

Put ` x = tan theta =>theta = tan^(-1) x`

`:. tan^(-1)  (sqrt(1+x^2) - 1)/x = tan^(-1)  ((sqrt(1 + tan^2 theta) - 1)/ tan theta)`

`= tan^(-1) ((sec theta -1)/tan 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) = theta/2 = 1/2 tan^(-1) x`

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APPEARS IN

NCERT Class 12 Maths
Chapter 2 Inverse Trigonometric Functions
Q 5 | Page 47
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