If `sin^-1x+sin^-1y+sin^-1z=(3pi)/2 and f(1)=2. f(p+q)=f(p). f(q)AAp,qinR.,` then `x^(f(1))+y^(f(2))+z^(f(3))-(x+y+z)/(x^(f(1))+y^(f(2))+z^(f(3))` is equal to

Options

A. 0
B. 1
C. 2
D. 3

Explanation

We know that
`-(pi)/(2)lesin^(-1)xsin^(-1)ysin^(-1)zle(pi)/(2)`
`therefore sin^(-1)x+sin^(-1)y+sin^(-1)z=(3pi)/(2)`
`rarr sin^(-1)x=sin^(-1)y=sin^(-1)z=(pi)/(2)`
`x=y=z=1`
`f(x+y)=f(x)f(y)` for all x,y in R
`therefore f(x)=2^(x)` for all x in R
`rarr f(1)=2,f(2)=4 and f(3)=8`
`therefore x^(f(1))+y^(f(2))+z^(f(3))-(x+y+z)/(x^(f(1))+y^(f(2))+z^(f(3))`
`=1+1+1-(1+1+1)/(1+1+1)=3-1=2`