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If 0ltxlt1 then sqrt(1+x^(2))[{x cos (c...

If `0ltxlt1` then `sqrt(1+x^(2))[{x cos (cot^(-1)x)+sin(cot^(-1)x}^(2)-1]^(1/2)`

A

`(x)/sqrt(1+x^(2))`

B

x

C

`xsqrt(1+x^(2))`

D

`sqrt(1+x^(2))`

Text Solution

AI Generated Solution

To solve the given expression \( \sqrt{1+x^2} \left[ x \cos(\cot^{-1} x) + \sin(\cot^{-1} x) \right]^2 - 1 \), we will follow these steps: ### Step-by-Step Solution: 1. **Let \( \theta = \cot^{-1}(x) \)**: - By definition, \( \cot(\theta) = x \). - This implies that in a right triangle, the adjacent side is \( x \) and the opposite side is \( 1 \). ...
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Knowledge Check

  • If 0 lt x lt 1", then " sqrt(1+x^(2))[{x cos (cot^(-1)x) + sin ( cot^(-1) x)}^(2) -1]^(1//2) is equal to

    A
    `x/sqrt(1+x^(2))`
    B
    `x`
    C
    `xsqrt(1+x^(2))`
    D
    `sqrt(1+x^(2))`
  • If 0lt xlt1, then sqrt(1+x^2)[{x cos (cot^-1x)+sin(cot^-1x)}^2-1]^(1//2) is equal to

    A
    `x/sqrt(1+x^2)`
    B
    x
    C
    `xsqrt(1+x^2)`
    D
    `sqrt(1+x^2)`
  • If 0 lt x le 1, " then " sqrt(1+x^2) {x cos (cot^(-1)x)+sin(cot^(-1)x)}^(1//2)=

    A
    `x/(sqrt(1+x^2))`
    B
    x
    C
    `x sqrt(1+x^2)`
    D
    none of these
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    cot^(-1)(sqrt(1+x^2)-x)

    cot^(-1)((sqrt(1-x^(2)))/(x))

    if 0 lt x lt 1 , then sqrt( 1+ x^(2))[ { x cos ( cot^(-1) x ) + sin ( cot^(-1) x)} ^(2) -1]^(1//2) is equal to

    Statement 1: If x=(1)/(5 sqrt(2)) , then [x cos(cot^(-1)x)+sin(cot^(-1)x)]^(2)=(51)/(50) . Statement 2: tan["cot"^(-1)(1)/(5sqrt(2))-"sin"^(-1)(4)/(sqrt(17))]=(29)/(3) .

    cot^(-1)((sqrt(1+x^(2))-1)/(x)) =