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

Prove that: `tan^(-1)sqrt(x)=1/2cos^(-1)((1-x)/(1+x)), x in [0,1]`

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Let `sqrt(x)=tan thetaimpliestan^(2)theta`
`:. RHS=(1)/(2) cos^(-1)""((1-x)/(1+x)), " " x in [0,1]`
`=(1)/(2) cos^(-1)""((1-tan^(2) theta)/(1+tan^(2)theta))`
`=(1)/(2)cos^(-1)(cos2theta)=(1)/(2)(2theta)`
`=theta=tan^(-1)sqrt(x)=` LHS Hence Proved.
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ALLEN -INVERSE TRIGONOMETRIC FUNCTIONS-All Questions
  1. Evaluate: tan{1/2cos^(-1)(sqrt5/3)}

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  2. Prove that: 2tan^(-1)1/2+tan^(-1)1/7=tan^(-1)(31)/(17)

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  3. Prove that: tan^(-1)sqrt(x)=1/2cos^(-1)((1-x)/(1+x)), x in [0,1]

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  4. Find the value of 4tan^(-1)(1/5)-tan^(-1)(1/(70))+tan^(-1)(1/(99))

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  5. Prove that :2tan^(-1)(1/5)+sec^(-1)((5sqrt(2))/7)+2tan^(-1)\ 1/8=pi/4

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  6. The equation 2cos^(-1)x+sin^(-1)x=(11pi)/6 has a. no solution b. only...

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  7. Solve the equation tan^(-1) ((x-1)/(x-2))+tan^(-1)((x+1)/(x+2))=pi/4

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  8. Solve the equation x^(2)+(x/(x-1))^(2)=8

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  9. Find the complete solution set of sin^(-1)(sin5)> x^2-4x

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  10. The complete solution set of the inequality [cot^(-1)x]^2-6[cot^(-1)x]...

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

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  12. If tan^(-1)y=4tan^(-1)x, (|x| lt tan(pi) /8) , find y as an algebraic...

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  13. If A=2tan^(-1)(2sqrt(2)-1) and B=3sin^(-1)(1/3)+sin^(-1)(3/5), then wh...

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  14. If [sin^-1 (cos^-1(sin^-1 (tan^-1 x)))]=1, where [*] denotes the gre...

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  15. Transform sin^(-1)x in other inverse trigonometric function.

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  16. The value of sin^(-1)(-sqrt(3)/2)\ is: (A) (-pi)/3 (B) (-2pi)/3 (...

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  17. cos(2 tan^(-1)(1/7))equals: a."sin"(4cot^(-1)3) b. "sin"(3cot^(-1)4)...

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  18. The value of sec(sin^-1sin((-50pi)/9)+cos^-1cos((31pi)/9))

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  19. cos (cos^-1 cos((8pi)/7) +tan^-1 tan((8pi)/7)) has the value equal to ...

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  20. (sin^(-1) x)^2 + (sin^(-1) y)^2 + 2(sin^(-1) x)(sin^(-1) y) = pi^2 , t...

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