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If -1 lt x lt 0, then cos^(-1) x is equa...

If `-1 lt x lt 0`, then `cos^(-1) x` is equal to

A

`sec^(-1).(1)/(x)`

B

`pi - sin^(-1) sqrt(1 -x^(2))`

C

`pi + tan^(-1).(sqrt(1 -x^(2)))/(x)`

D

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

Text Solution

Verified by Experts

The correct Answer is:
A, B, C, D
`cos^(-1) x = sec^(-1).(1)(x) " for all " x in [-1, 0)`
Leet `x = -y`
`:. Cos^(-1) x = cos^(-1) (-y) = pi - cos^(-1) y`
`= pi - sin^(-1) sqrt(1 -y^(2))`...(i)
`=pi - tan^(-1).(sqrt(1 - y^(2)))/(y)` ..(ii)
From (i), `cos^(-1) = pi - sin^(-1) sqrt(1 - y^(2)) = pi - sin^(-1) sqrt(1 -x^(2))`
From (ii), `cos^(-1) x = pi - tan^(-1).(sqrt(1 -y^(2)))/(y)`
`= pi - tan^(-1).(sqrt(1 -x^(2)))/(-x)`
`= pi + tan^(-1).(sqrt(1 -x^(2)))/(x)`
`= pi + cot^(-1).(x)/(sqrt(1 -x^(2))) -pi " " ("as " x lt 0)`
`= cot^(-1).(x)/(sqrt(1 -x^(2)))`
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CENGAGE-INVERSE TRIGONOMETRIC FUNCTIONS-Exercise (Multiple)
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  7. If z = sec^(-1) (x + 1/x) + sec^(-1) (y + 1/y), where xy< 0, then the ...

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  8. The value of k(k >0) such that the length of the longest interval in w...

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  9. Which of the following pairs of function/functions has same graph? y=...

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  10. If sin^(-1) x + sin^(-1) y = (pi)/(2) and sin 2x = cos 2y, then

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  11. If cos^(-1)x + cos^(-1)y + cos^(-1)z = 3pi, then xy + yz +zx is equal ...

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  12. If sin^(-1)(a-a^2/3+a^3/9-...)+cos^(-1)(1+b+b^2+...)=pi/2 then find a ...

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  13. If tan^(-1)(x^2+3|x|-4)+cot^(-1)(4pi+sin^(-1)s in 14)=pi/2, t h e n th...

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  14. If 2tan^(-1)x+sin^(-1)((2x)/(1+x^2) ) is independent of x then :

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  15. If alpha=tan^(-1)((4x-4x^3)/(1-6x^2+x^2)),beta=2sin^(-1)((2x)/(1+x^2))...

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  16. 2"tan"(tan^(-1)(x)+tan^(-1)(x^3)),w h e r ex in R-{-1,1}, is equal to...

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  17. Let alpha=som^(-1)((36)/(85)),beta=cos^(-1)(4/5)a n dgamma=tan^(-1)(8/...

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  18. If Sn=cot^-1(3)+cot^-1(7)+cot^-1(13)+cot^-1(21)+....., n terms, then

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  19. Equation 1+x^2+2x"sin"(cos^(-1)y)=0 is satisfied by exactly one value ...

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