Consider the following statements : 1. `sin^(-1)\ 4/5 + sin^(-1)\ 3/5 =pi/2` and 2. `tan^(-1)sqrt3+tan^(-1) 1= -tan^(-1) (2+sqrt3)` Which of the above statement(s) is/are correct?

Options

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

Answer

A

Explanation

`sin^(-1)x+sin^(-1)y=sin^(-1)(x sqrt(1-y^(2))+y sqrt(1-x^(2)))`
If `(-1 le x, y le 1)` & `(x^(2)+y^(2) le 1)`
`rArr sin^(-1)((4)/(5))+"sin"^(-1)(3)/(5)`
`=sin^(-1)[(4)/(5)sqrt(1-((3)/(5))^(2))+(3)/(5)sqrt(1-((4)/(5))^(2))]`
`=sin^(-1)[(16)/(25)+(9)/(25)]=sin^(-1)(1)=(pi)/(2)`
`therefore` Statement (1) is correct
Again, `tan^(-1)x + tan^(-1)y = pi + tan^(-1)[(x+y)/(1-xy)]`
If, `(x gt 0), (y gt 0) and (xy gt 1)`
`tan^(-1)(sqrt(3))+ tan^(-1)(1)=pi + tan^(-1)[(sqrt(3)+1)/(1-sqrt(3))]`
`=pi+tan^(-1)[((sqrt(3)+1)(1+sqrt(3)))/((1-sqrt(3))(1+sqrt(3)))]`
`=pi + tan^(-1)((4+2sqrt(3))/(-2))=pi + tan^(-1)[-(2+sqrt(3))]`
`=pi - tan^(-1)(2 + sqrt(3))" "because tan^(-1)(-x)=-tan^(-1)x`