Simplify (i) `sin(pitan{cot^(-1)((-2)/3)})`
(ii) `sin(2"tan"^(-1)1/2)`
(iii) `cos(2cos^(-1)(1//5)+sin^(-1)(1//5))`

Explanation

(i) Let `y=tan{cot^(-1)((-2)/3)}`……………….(A)
`:'cot^(-1)(-x)=pi-cot^(-1)x,xepsilonR`
(A) can be written as
`y=tan{pi-cot^(-1)(2/3)}`
`y=-tan("cot"^(-1)2/3)`
`:'cot^(-1)x="tan"^(-1)1/x` if `xgt0`
`:.y=-tan("tan"^(-1)3/2)impliesy=-3/2` so `sin(pitan{"cot"^(-1)((-2)/3)})=sin(-(3pi)/2)=1`
(ii) `sin(2"tan"^(-1)1/2)=2sin("tan"^(-1)1/2)cos("tan"^(-1)1/2)=sin("sin"^(-1)1/(sqrt(5)))xxcos("cos"^(-1)2/(sqrt(5)))`
`=2xx1/(sqrt(5))xx2/(sqrt(5))=4/5`
(iii) `cos(2"cos"^(-1)1/5+"sin"^(-1)1/5)=cos("cos"^(-1)1/5+"sin"^(-1)1/5+"cos"^(-1)1/5)`
`=cos((pi)/2+"cos"^(-1)1/5)=-sin (cos^(-1)(1/5))`..............i
`=-sqrt(1-(1/5)^(2))=-(2sqrt(6))/5`