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Write the Function in the Simplest Form: `Tan^(-1) (Sqrt(1+X^2) -1)/X, X != 0` - CBSE (Commerce) Class 12 - Mathematics

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Question

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

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 Solution for Mathematics Textbook for Class 12 (2018 to Current)
Chapter 2: Inverse Trigonometric Functions
Q: 5 | Page no. 47
Solution Write the Function in the Simplest Form: `Tan^(-1) (Sqrt(1+X^2) -1)/X, X != 0` Concept: Properties of Inverse Trigonometric Functions.
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