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tan ^(-1)x-tan ^(-1)y=sin ^(-1) ""(x-y)/...

`tan ^(-1)x-tan ^(-1)y=sin ^(-1) ""(x-y)/(sqrt((1+x^(2))(1+y^(2)))`

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prove that . Tan ^(-1) x- tan ^(-1 )y = cos ^(-1) (1+xy)/(sqrt((1+x^(2))(1+y^(2)))).

tan ^(-1)""(1-x)/(1+x)- tan ^(-1 )""(1-y)/(1+y) = sin ^(-1) ""(y-x)/(sqrt((1+x^(2))(1+y^(2)))

sin^(-1)x+sin^(-1)y=cos^(-1)""{sqrt((1-x^(2))(1-y^(2)))-xy}

tan ^(-1)""(1-x)/(1+x)""=(1)/(2) sin ^(-1)""(x)/(sqrt(1+x^(2)))

tan^(-1)x+tan^(-1) (1-x) = 2 tan ^(-1) sqrt(x(1-x))

Simplify y=tan^(-1)(x/(1+sqrt(1-x^2)))

Let tan ^(-1) y= tan ^(-1) x+ tan ^(-1) ""((2x)/(1-x^(2))) where |x| lt (1)/(sqrt(3)) . Then a value of y is -

tan^(-1)""(x-y)/(1+xy)+tan^(-1)""(y-z)/(1+yz)+tan^(-1)""(z-x)/(1+zx) =tan^(-1)""(x^(2)-y^(2))/(1+x^(2)y^(2))+tan^(-1)""(y^(2)-z^(2))/(1+y^(2)z^(2))+tan^(-1) ""(z^(2)-x^(2))/(1+z^(2)x^(2))

tan [(1)/(2) sin^(-1)""(2x)/(1+x^(2))+(1)/(2) cos ^(-1)""(1-y^(2))/(1+y^(2))], xy ne 1

tan (sin ^(-1)x+sin ^(-1) y) +tan (cos^(-1)x+cos ^(-1)y)

CHHAYA PUBLICATION-TRIGONOMETRIC INVERSE CIRCULAR FUNCTIONS-Short answer type questions
  1. tan^(-1)""(3)/(2) +tan^(-1)""(6)/(5)=pi - tan^(-1)""(27)/(8)

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  2. tan((1)/(2) sin ^(-1)""(2x)/(1+x^(2))+(1)/(2)cos^(-1)((1-x^(2))/(1+x^(...

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  3. tan ^(-1)x-tan ^(-1)y=sin ^(-1) ""(x-y)/(sqrt((1+x^(2))(1+y^(2)))

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  4. sin^(-1)x+sin^(-1)y=cos^(-1)""{sqrt((1-x^(2))(1-y^(2)))-xy}

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  5. sin^(-1) ""(2a)/(1+a^(2))-cos^(-1) ""(1-b^(2))/(1+b^(2))=2tan ^(-1) ""...

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  6. cot^(-1)""(xy+1)/(x-y)+cot^(-1)""(yz+1)/(y-z) +cot ^(-1) "" (zx +1)/(...

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  7. tan^(-1)""(x-y)/(1+xy)+tan^(-1)""(y-z)/(1+yz)+tan^(-1)""(z-x)/(1+zx) =...

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  8. cos^(-1)x= 2 sin ^(-1) sqrt((1-x)/(2))=2 cos ^(-1)""sqrt((1+x)/(2))=2t...

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  9. { cos (sin ^(-1) x)}^(2) ={sin (cos ^(-1) x)}^(2)

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  10. tan ^(1)(cotx) + cot ^(-1) (tan x)=pi -2x

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  11. tan^(-1) (cot x) +cot^(-1)(tan x) =pi -2x

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  12. sin cos ^(-1) tan (sec^(-1) x) = sqrt(2-x^(2))

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  13. sin "cosec"^(-1) cot (tan^(-1)x) =x

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  14. sin cot^(-1)cos (tan ^(-1)x)=sqrt((x^(2)+1)/(x^(2)+2))(x gt 0)

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  15. tan [(1)/(2) sin^(-1)""(2x)/(1+x^(2))+(1)/(2) cos ^(-1)""(1-y^(2))/(1...

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  16. cos (2cos ^(-1) x+sin ^(-1) x) , " when " x=(1)/(5) where 0 le ...

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  17. 2 tan ^(-1)""(1+x)/(1-x)+sin ^(-1)""(1-x^(2))/(1+x^(2))

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  18. tan (sin ^(-1)x+sin ^(-1) y) +tan (cos^(-1)x+cos ^(-1)y)

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  19. If xy=1 + a^(2) then show that tan^(-1) ""(1)/(a+x)+tan ^(-1) ...

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  20. "If " tan^(-1) x+ tan^(-1) y+tan ^(-1)z=(pi)/(2) , show that , x...

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