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Show that sin^(-1)(2x)/(1+x^(2))=[(2tan^...

Show that `sin^(-1)(2x)/(1+x^(2))=[(2tan^(-1)x,"if",|x|le1),(pi-2tan^(-1)x,"if",xgt1),(-(pi+2tan^(-1)x),"if",xlt-1):}`

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Given that , tan^(-1) ((2x)/(1-x^(2))) = {{:(2 tan^(-1) x"," |x| le 1),(-pi +2 tan^(-1)x","x gt 1),(pi+2 tan^(-1)x"," x lt -1):} sin^(-1)((2x)/(1+x^(2))) ={{:(2 tan^(-1)x","|x|le1),(pi -2 tan^(-1)x","x gt 1 and ),(-(pi+2tan^(-1))","x lt -1):} sin^(-1) x + cos^(-1) x = pi//2 " for " - 1 le x le 1 If cos^(-1). (6x)/(1 + 9x^(2)) = - pi/2 + 2 tan^(-1) 3x" , then " x in

Given that , tan^(-1) ((2x)/(1-x^(2))) = {{:(2 tan^(-1) x"," |x| le 1),(-pi +2 tan^(-1)x","x gt 1),(pi+2 tan^(-1)x"," x lt -1):} sin^(-1)((2x)/(1+x^(2))) ={{:(2 tan^(-1)x","|x|le1),(pi -2 tan^(-1)x","x gt 1 and ),(-(pi+2tan^(-1))","x lt -1):} sin^(-1) x + cos^(-1) x = pi//2 " for " - 1 le x le 1 sin^(-1) ((4x)/(x^(2)+4)) + 2 tan^(-1)( - x/2) is independent of x , then

If tan^(-1)((1-x)/(1+x))=1/2tan^(-1)x,xgt0 find x

2 tan^(-1) x = sin^(-1) ((2x)/(1+x^(2))) , 1 le x le 1

tan ^(-1) ""(x-1)/(x-2) + tan ^(-1) ""(x+1)/(x+2)=(pi)/(4)

Show that cot^(-1)x={(tan^(-1)(1//x),xgt0),(pi+tan^(-1)(1//x),xlt0):}

Show that cot^(-1)x={(tan^(-1)(1//x),xgt0),(pi+tan^(-1)(1//x),xlt0):}