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Prove that : tan^(-1)((sqrt(1+x^(2))+s...

Prove that :
`tan^(-1)((sqrt(1+x^(2))+sqrt(1-x^(2)))/(sqrt(1+x^(2))-sqrt(1-x^(2))))=pi/4+1/2cos^(-1)x^(2)`.

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Verified by Experts

The correct Answer is:
`(-x)/(sqrt(1-x^(4)))`
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KUMAR PRAKASHAN-CONTINUITY AND DIFFERENTIABILITY-NCERT Exemplar Problems and Solution (Short Answer Type Questions)
  1. Differentiate sec^(-1) ((1)/(4x^(3)-3x)), 0 lt x lt (1)/(sqrt2)

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  2. Write the following functions in the simplest form : tan^(-1)((3a^(2...

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  3. Prove that : tan^(-1)((sqrt(1+x^(2))+sqrt(1-x^(2)))/(sqrt(1+x^(2))-s...

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  4. Find (dy)/(dx): x=t + (1)/(t) and y= t - (1)/(t)

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  5. Find (dy)/(dx) of each of the functions expressed in parametric form: ...

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  6. Find (dy)/(dx) of each of the functions expressed in parametric form: ...

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  7. Find (dy)/(dx) of each of the functions expressed in parametric form: ...

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  8. Find (dy)/(dx) of each of the functions expressed in parametric form: ...

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  9. Find (dy)/(dx): x= e^(cos 2t) and y= e^(sin 2t) show that, (dy)/(dx)...

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  10. Find (dy)/(dx): x=a sin 2t (1+ cos 2t) and y= b cos 2t (1-cos 2t) ...

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  11. If x= 3 sin t- sin (3t), y= 3 cos t- cos 3t, then find ((dy)/(dx)) " ...

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  12. Differentiate (x)/(sin x) w.r.t sin x

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  13. Differentiate tan^(-1) ((sqrt(1+x^(2))-1)/(x)) w.r.t tan^(-1)x, where ...

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  14. Find (dy)/(dx) when x and y are connected by the relation given: si...

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  15. Find (dy)/(dx) when x and y are connected by the relation given: se...

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  16. Find (dy)/(dx) when x and y are connected by the relation given: ta...

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  17. Find (dy)/(dx) when x and y are connected by the relation given: (x...

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  18. If ax^(2) + 2hxy + by^(2) + 2gx + 2f y + c = 0, then show that (dy)/(d...

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  19. If x= e^((x)/(y)), then prove that (dy)/(dx)= (x-y)/(x.log x)

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  20. If y^(x)= e^(y-x), then prove that (dy)/(dx)= ((1+ log y)^(2))/(log y)

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