If ` tan x = (2t)/(1 -t^(2)) " and sin y" = (2t)/(1 + t^(2))` , then the value of
` (dy)/(dx)` is

Options

A. 1
B. t
C. ` ( 1 ) /( 1 - t ) `
D. ` ( 1 ) /( 1 + t ) `

Answer

A

Explanation

Given, ` tan x = ( 2t )/( 1- t ^ 2 ) and sin y = ( 2t ) /( 1 + t ^ 2 ) `
Now,` x =tan ^( -1 ) ( ( 2t ) /( 1- t^ 2)) `
` x = 2 tan ^( - 1 ) t " " `… (i)
and ` y = sin ^( - 1 ) ((2t ) /( 1 + t^2 )) `
` y = 2 tan ^( -1 ) t " " `... (ii)
From Eq. (i)
` (dx )/ ( dt ) = ( 2 )/( 1 +t ^ 2 ) `
From Eq. (ii)
` ( dy )/( dt ) = ( 2 )/( 1 + t^ 2 ) `
` therefore (dy )/(dx) = ( dy ) /(dt ) xx ( dt ) /( dx ) = ( 2 ) /( (1 + t^ 2 )) xx (( 1 + t^ 2 ))/( 2 ) = 1 `