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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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We have to prove

`\tan ^{-1} \sqrt{x}=\frac{1}{2} \cos ^{-1}(\frac{1-x}{1+x}), x \in(0,1)`

L.H.S.

`=\tan ^{-1} \sqrt{x}=\frac{1}{2}[2 \tan ^{-1} \sqrt{x}]` ...
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RD SHARMA-INVERSE TRIGONOMETRIC FUNCTION-Solved Examples And Exercises
  1. Show that 2tan^-1 (-3)=-pi/2+tan^-1(-4/3)

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  2. Show that : cos(2\ tan^(-1)(1/7))=sin(4\ tan^(-1)(1/3))

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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 expression: sin(2tan^(-1)(1/3))+cos(tan^(-1)2sqrt(2)...

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  5. Differentiate tan^(-1)((3a^2x-x^3)/(a^3-3a x^2)),\ -1/(sqrt(3))<x/a<1/...

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  6. 4tan^(- 1)(1/5)=tan^(- 1)(1/70)+tan^(- 1)(1/99)+pi/4

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

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  8. Evaluate: tan{2t a n^-1 (1/5)-pi/4}

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  9. Solve for x : 2tan^(-1)(cosx)=tan^(-1)(2cos e cx)

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  10. Solve sin^(-1)(1-x)-2s in^(-1)x=pi/2

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  11. Solve: sin[2\ cos^(-1){cot(2\ tan^(-1)x)}]=0

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  12. Solve the following equation: sin^(-1)x+sin^(-1)(1-x)=cos^(-1)x

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  13. Solve for x : tan^(-1)((2x)/(1-x^2))+cot^(-1)((1-x^2)/(2x))=pi/3,-1ltx...

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  14. If cos^-1 (x^2-1)/(x^2+1)+ tan^-1 (2x)/(x^2-1) = (2pi)/3, then x equal...

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  15. If y=cot^(-1)(sqrt(cos"x"))-tan^(-1)(sqrt(cos"x")), prove that siny=ta...

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  16. Prove that : cos^(-1)((cosalpha+cosbeta)/(1+cosalphacosbeta))=2\ tan^(...

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  17. Show taht 2tan^-1(tan(alpha/2)tan(pi/4-beta/2))=tan^-1((sinalphacosbet...

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  18. Prove that: (alpha^3)/2cos e c^2(1/2tan^(-1)(alpha/beta))+(beta^3)/2s ...

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  19. Prove that : tan^(-1){(cos2\ alphasec2\ beta+cos2betasec2alpha)/2}=tan...

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  20. Evaluate the following: (i) tan{2\ tan^(-1)(1/5)-pi/4} (ii) tan(1/2...

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