Find the following values :
(a) `tan^(1) tan.(13pi)/(5)` (b) `sec^(-1) sec.(13pi)/(3)`
(c) `sin^(-1) sin.(33pi)/(5)` (d) `sin^(-1) (sin 8)`
(e) `tan^(-1) (tan 10)` (f) `sec^(-1) (sec 9)`
(g) `cot^(-1) (cot 6)` (h) `cosec^(-1) (cosec 7)`

Answer

(a) `-(2pi)/(5)` (b) `(pi)/(3)` (c) `(2pi)/(5)` (d) `3 pi -8` (e) `10 - 3 pi` (f) `9 - 2pi` (g) `6 - pi` (h) `7 - 2pi`

Explanation

(a) `tan^(-1) tan. (13 pi)/(5) = tan^(-1) tan {3pi + (-(2pi)/(5))}`
`= tan^(-1) tan.(-(2pi)/(5))`
`= -(2pi)/(5)`
(b) `sec^(-1) sec. (13pi)/(3) = sec^(-1) sec.(3pi + (pi)/(3))`
`= sec^(-1) sec. (pi)/(3)`
`= (pi)/(3)`
(c) `sin^(-1) sin.(33 pi)/(5) = sin^(-1) sin. (7 pi - (2pi)/(5))`
`= sin^(-1) sin. (2pi)/(5) = (2pi)/(5)`
(d) `sin^(-1) (sin 8) = sin^(-1) (sin (3pi -8)) = 3 pi -8`
(e) `tan^(-1) (tan 10) = tan^(-1) (tan(10 - 3 pi)) = 10 - 3 pi`
(f) `sec^(-1) (sec 9) = sec^(-1) (sec(9 - 2pi)) = 9 - 2pi`
(g) `cot^(-1) (cot 6) = cot^(-1) (cot(6 - pi)) = 6 - pi`
(h) `cosec^(-1) (cosec7) = cosec^(-1) (cosec(7 - 2 pi)) = 7 - 2pi`