Evaluate:
i) `tan{1/2cos^(-1)sqrt(5)/3}`,
ii) `tan{2tan^(-1)1/5-pi/4}`

Explanation

i) Let `cos^(-1)sqrt(5)/3=theta`. Then, `costheta=sqrt(5)/3`.
`therefore tan{1/2cos^(-1)sqrt(5)/3}=tantheta/2`
`=sqrt((1-costheta)/(1+costheta))=sqrt((1-sqrt(5)/3)/(1+sqrt(5)/3))=sqrt((3-sqrt(5))/(3+sqrt(5))`
`=sqrt((3-sqrt(5))/(3+sqrt(5)) xx (3-sqrt(5))/(3+sqrt(5)) = (3-sqrt(5))/sqrt(9-5)`
`=(3-sqrt(5))/sqrt(4)=(3-sqrt(5))/(2)`
ii) `tan{2tan^(-1)1/5-pi/4}`
`=tan{tan^(-1)(2 xx 1/5)/(1-1/25))-tan^(-1)1}`
`=tan{tan^(-1)(2/5 xx 25/24)-tan^(-1)1}`.
`=tan{tan^(-1)5/12-tan^(-1)1}`
`=tan{tan^(-1)(5/12-1)/(1+5/12)}=tan{tan^(-1)(-7/17)}=-7/17`.