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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 ENGLISH-INVERSE TRIGONOMETRIC FUNCTIONS -Section I - Solved Mcqs
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  2. The set of values of x for which tan^(-1)(x)/sqrt(1-x^(2))=sin^(-1) ...

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

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  7. Solve sin^(-1)[((2x^2+4)/(1+x^2))]<pi-3

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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-ddot)+cos^(-1)(x^2-(x^4)/2+(x^6)/4)=pi/2...

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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. Find the values of sin (cos−1 3/5)

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

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

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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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