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while differentiating one inverse trigonometric function with respect to another trigonometric function ,we take "x=tan Y" OR :"X=COS Y" OR "X=SIN Y" after the substitution of the x we may get "tan inverse of tan 2y " for some "f(x)" after with out checking any conditions we write "f(x)" as "2y" why are we not checking the conditions in all cases????? 10 years ago

dear nikhil , i m explaning this with the help of example ...

suppose we have to diffenertiate sin-1x wrt cos-1x ....

this means  , methametically this can be written as dsin-1x/dcos-1x

now  , let sin-1x = Z & cos-1x = Y

Z = sin-1x

dZ/dx = d/dx (sin-1x) = 1/(1-x2)1/2            .................1

Y = cos-1x

dY/dx = -1/(1-x2)1/2               .....................2

dividing  1 by  2

dZ/dY = -1

now , Z = sin-1x & Y = cos-1x so

differentiation of sin-1x wrt cos-1x is -1 ...

now see another example ,

differentiate sin-1x wrt tan-1x ...

let Z = sin-1x  & Y = tan-1x then

dZ/dx = 1/(1-x2)1/2 &             ............1

dY/dx = 1/1+x2                 ................2

divide 1 by 2

dZ/dY = 1+x2/(1-x2)1/2 = differentiation of sin-1x wrt tan-1x ...