Which of the following pairs of function/functions has same graph? `y=tan(cos^(-1)x); y=(sqrt(1-x^2))/x` `y=t a n(cot^(-1)x);y=1/x` `y="sin"(tan^(-1)x); y=x/(sqrt(1+x^2))` `y="cos"(tan^(-1)x); y=sin(cot^(-1)x)`

Options

A. `y = tan (cos^(-1) x), y = (sqrt(1 - x^(2)))/(x)`
B. `y = tan (cot^(-1) x), y = (1)/(x)`
C. `y = sin (tan^(-1) x), y = (x)/(sqrt(1 + x^(2)))`
D. `y = cos (tan^(-1) x), y = sin (cot^(-1) x)`

Answer

A, B, C, D

Explanation

(1) `y = tan (cos^(-1)x) and y = (sqrt(1 -x^(2)))/(x)`
or `y = tan (tan^(-1).(sqrt(1 -x^(2)))/(x)) = (sqrt(1 -x^(2)))/(x) and y = (sqrt(1 -x^(2)))/(x)`
`:. D_(1) = [-1, 1] - {0} and D_(2) = [-1, 1] {0}`
So functions are identical and hence they have same graph
(2) `y = tan (cot^(-1) x) and y = (1)/(x)`
`:. y = tan (tan^(-1).(1)/(x)) adn y = (1)/(x)`
or `y = (1)/(x) adn y = (1)/(x)`
`:. D_(1) = R - {0} and D_(2) = R - {0}`
So, functions are identical and hence they have same graph
(3) `y = sin (tan^(-1) x) and y = (x)/(sqrt(1 + x^(2)))`
or `y = sin (sin^(-1).(x)/(sqrt(1 + x^(2)))) = (x)/(sqrt(1 + x^(2))) and y = (x)/(sqrt(1 + x^(2)))`
or `D_(1) = R, D_(2) = R`
So, functions are identical and hence they have same graph.
(4) `y = cos (tan^(-1) x) and y = sin (cot^(-1) x)`
or `y = cos (tan^(-1) x) = cos [cos^(-1).(1)/(sqrt(1 + x^(2)))] = (1)/(sqrt(1 + x^(2)))`
and `y = sin (cot^(-1) x) = sin (sin^(-1).(1)/(sqrt(1 + x^(2)))) = (1)/(sqrt(1 + x^(2)))`
`:. D_(1) = R and D_(2) = R`
So, functions are identical and hence they have same graph.