Question

Statement-1: If `alpha,beta` roots of the equation `18(tan^(-1)x)^(2)-9pi tan^(-1)x+pi^(2)=0 then alpha+beta =(4)/sqrt(3)` Statement-2: `sec^(2)cos^(-1)(1/4)+cosec^(2)sin^(-1)(1/5)=41`
A. Statement-1 is is True, Statement-2 is true, Statement-2 is a correct explanation for Statement-1.
B. Statement-1 is True, Statement-2 is True, Statement-2 is not a correct explanation for Statement-1.
C. Statement-1 is True, Statement-2 is False.
D. Statement-1 is False, Statement-2 is True.

Answer 1

we have `(tan^(-1)x)^(2)-(pi)/(2)tan^(-1)x+(pi)^(2)/(18)=0`
`rarr (tan^(-1)x)^(2)-((pi)/(6)+(pi)/(3))tan^(-1)x+(pi^(2))/(18)=0 `
`rarr tan^(-1)x=(pi)/(6),(pi)/(3) rarr tan^(-1) alpha =(pi)/(6) and tan^(-1) beta =-(pi)/(3)`
`rarr alpha = tan (pi)/(6) =(1)sqrt(3) and beta =tan (pi)/(3)=sqrt(3) rarr alpha+beta =(4)/sqrt(3)`
`so statement 1 is true
`sec^(2)(cos^(-1)1/4)+cosec^(2)(sin^(-1)1/5)`
`={sec(sec^(-1)4)}^(2)+(cosec^(-1)5)}^(2)=16+25=4`
so statement 2 is true