Match the following lists:

Answer

`a to r,s; b to p,q,r,s; c to s; d to p`

Explanation

a. `tan^(-1)((2x)/(1-x^(2))) in (-(pi)/(2),(pi)/(2))`
`or 2 tan^(-1)x in (-(pi)/(2),(pi)/(2))`
`or tan^(-1)x in (-(pi)/(4),(pi)/(4))`
`or tan^(-1)x in (-1,1)`
b. `f(x)=sin^(-1)(sinx) and g(x)=sin(sin^(-1)x)`
`f(x)` is defined if `sin x in [-1,1]` which is true for all `x in R`.
But g(x) is defined for only `x in [-1,1]`.
Hence, `f(x)` and g(x) are identical if `x in [-1,1]`.
c. `f(x)=log_(x^(2))25 and g(x)=log_(x)5`
`f(x)` is defined ` AA x in R - {0,1} and g(x) ` is defined for `(0, oo) -{1}.`
Hence, `f(x)` and g(x) are identical if `x in (0,1) cup (1,oo).`
d. `f(x)=sec^(-1)x+"cosec"^(-1)x,g(x)=sin^(-1)x+cos^(-1)x`
`f(x)` has domain `R -(-1,1) and g(x)` has domain [-1, 1]
Hence, both the functions are identical only if `x=-1,1.`