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

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

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OMEGA PUBLICATION-INVERSE TRIGONOMETRIC FUNCTIONS -Important Questions from Miscellaneous Exercise
  1. Prove that tan^(-1)""(1)/(2)+tan^(-1)""(2)/(11)=tan^(-1)""(3)/(4)

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  2. Prove that tan^(-1)""(2)/(11)+tan^(-1)""(7)/(24)=tan^(-1)""(1)/(2).

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  3. Express tan^(-1)((cosx)/(1-sinx)),-(pi)/(2)ltxlt(pi)/(2) in the simple...

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  4. Prove that tan^(-1) (1/5)+ tan^(-1) (1/7) + tan^(-1) (1/3) + tan^(-1) ...

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  5. Show that sin^(-1)""(3)/(5)-sin^(-1)""(8)/(17)=cos^(-1)""(84)/(85)

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  6. Prove that : cot^-1[(sqrt(1+sin x) + sqrt(1-sin x))/(sqrt1+sin x + sqr...

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  7. Prove that (9pi)/(8)-9/(4)"sin"^(-1)1/(3)=9/(4)"sin"^(-1)(2sqrt(2))/(3...

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  8. Solve tan^(-1)(x+1)+tan^(-1)(x-1)=tan^(-1)""(8)/(31)

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

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  10. Solve the 2 tan^-1 (cos x) = tan^-1 (2 cosec x).

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  11. Prove that 3cos^(-1)x=cos^(-1)(4x^(3)-3x),x in[(1)/(2),1]

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  12. Show that "sin"^(-1)(5/(13))+"cos"^(-1)(3/(5))="tan"^(-1)(63/(16)).

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  13. Prove that : "tan"^(-1)(x/(4)+1/(2)"cos"^(-1)a/(b))+"tan"(pi/(4)-1/(2)...

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  14. Prove that : "sin"^(-1)(2xsqrt(1-x^(2)))=2"cos"^(-1)x,1/(sqrt(2))lexle...

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  15. Write in simplest form : "tan"^(-1)(sqrt((1-"cos"x)/(1+"cos"x))),xltpi...

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  16. Express "tan"^(-1)(("cos"x-"sin"x)/("cos"x+"sin"x)),0ltxltpi in the si...

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  17. Prove that : "tan"^(-1)(1/(2))+"tan"^(-1)(1/(5))+"tan"^(-1)(1/(8))=p...

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  18. Prove the following : "cos"^(-1)(4/(5))+"cos"^(-1)(12/(13))="cos"^(-...

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  19. Express "tan"^(-1)x/(sqrt(a^(2)-x^(2))),|x|lta, in the simplest form.

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  20. Express "tan"^(-1)(sqrt(1+x^(2))-1)/(x),xne0 in simplest form.

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