Consider the following statements :
1. There exists `theta in(-(pi)/(2),(pi)/(2))` for which `tan^(-1)(tan theta) ne theta`.
2. `sin^(-1)((1)/(3))-sin^(-1)((1)/(5))=sin^(-1)((2 sqrt(2)(sqrt(3)-1))/(15))`
Which of the above statements is/are correct?

Options

A. 1 only
B. 2 only
C. Both 1 and 2
D. Neither 1 nor 2

Answer

B

Explanation

`theta in (-(pi)/(2),(pi)/(2))` for which `tan^(-1)(tan theta)=theta`
Hence, statement (1) is incorrect.
if `x le 1, y le 1` & `x^(2) + y^(2) le 1`
`therefore sin^(-2)(x)-sin^(-1)y=sin^(-1)[x sqrt(1-y^(2))-y sqrt(1-x^(2))]`
`rArr "sin"^(-1)(1)/(3)-"sin"^(-1)(1)/(5)`
`=sin^(-1)[(1)/(3)sqrt(1-(1)/(25))-(1)/(5)sqrt(1-(1)/(9))]`
`=sin^(-1)[(1)/(3xx5) sqrt(24)-(1)/(5xx3)sqrt(8)]`
`=sin^(-1)[(sqrt(8xx3)-sqrt(8))/(15)]=sin^(-1)[(2sqrt(2)(sqrt(3)-1))/(15)]`
Hence, statement (2) is correct.