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cos^(-1){1/2x^(2)+sqrt(1+x^(2))sqrt(1-x^...

`cos^(-1){1/2x^(2)+sqrt(1+x^(2))sqrt(1-x^(2))/(4)}=cos^(-1)(x/2)-cos^(-1)x`

A

`x in [-1,1]`

B

`x in R`

C

`x in [0,1]`

D

`x in [-1,0]`

Text Solution

Verified by Experts

We observe that
`(x^(2))/(2)+sqrt(1-x^(2))sqrt(1-(x^(2))/(4))` is positive and defined for all
`therefore cos^(-1){(x)^(2)/(2)+sqrt(1-x^(2))sqrt(1-(x^(2))/(4)}in[0,(pi)/(2))]`
Now RHS of the given relatio is defined for
Aslo LHS`ge0`
`rarr cos^(-1)(x)/(2)-cos(-1)xge0rarrcos^(-1)(x)/(2)gecos^(-1)xrarrx in [0,1]`
Hence LHS=RHS for `x in[0,1]`
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OBJECTIVE RD SHARMA-INVERSE TRIGONOMETRIC FUNCTIONS -Section I - Solved Mcqs
  1. A root of the equation17x^2 + 17x tan [2 tan^-1(1/5) - pi/4] - 10 = 0 ...

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  2. The set of vlues of x for which tan^(-1)(x)/sqrt(1-x^(2))=sin^(-1) x...

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  3. cos^(-1){1/2x^(2)+sqrt(1+x^(2))sqrt(1-x^(2))/(4)}=cos^(-1)(x/2)-cos^(-...

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  4. If x in[-1/2,1] then sin^(-1)(sqrt(3)/(2)x-1/2sqrt(1-x^(2)))

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  5. cosec^(-1)(cosx) is defined if

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  6. If 0< x< 1,then tan^(-1)(sqrt(1-x^2)/(1+x)) is equal to

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  7. sin^(-1){sin(2x^(2)+4)/(x^(2)+1)=ltpi-3 if

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  8. cos[tan^(-1){tan((15pi)/4)}]

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  9. If Sigma(r=1)^(n) cos^(-1)x(r)=0, then Sigma(r=1)^(n) x(r) equals

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  10. If Sigma(r=1)^(2n) sin^(-1) x^(r )=n pi, then Sigma(r=1)^(2n) x(r )...

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  11. If sin^(- 1)(x-(x^2)/2+(x^3)/4-.....)+cos^(- 1)(x^2-(x^4)/2+(x^6)/4-....

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  12. If sin^(-1)sqrt(x^2+2x + 1) + sec^(-1)sqrt(x^2 + 2x + 1) = pi/2; x!= 0...

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  13. if cos^-1 x > sin^-1 x, then x belongs to the interval

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  14. The set of values of x, satisfying the equation tan^2(sin^-1x) > 1 is ...

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  15. If cot^(-1)(n/(pi))>(pi)/6, n in N, then the maximum value of n is :

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  16. if 6sin^-1(x^2-6x+12)=2pi, then the value of x, is

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  17. Which of the following is the solution set of the equation sin^-1 x = ...

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  18. If log(2)xge0 then log(1//pi){sin^(-1)((2x)/(1+x^(2)))+2tan^(-1)x} is...

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  19. The value of a for which a x^2+sin^(-1)(x^2-2x+2)+cos^(-1)(x^2-2x+2)=1...

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  20. The value of x that satisfies tan^(-1)(tan -3)=tan^(2)x is

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