Which of the following quantities is/are positive? `"cos"(tan^(-1)(t a n4))` (b) `"sin"(cot^(-1)(cot4))` `"tan"(cos^(-1)(cos5))` (d) `"cot"(sin^(-1)(sin4))`

Options

A. `cos (tan^(-1) (tan 4))`
B. `sin(cot^(-1)(cot 4))`
C. `tan (cos^(-1) (cos 5))`
D. `cot(sin^(-1) (sin 4))`

Answer

A, B, C

Explanation

(1) `cos (tan^(-1) (tan (4 - pi)))`
`= cos(4 - pi) = cos (pi -4) = - cos 4 gt 0`
(2) `sin(cot^(-1) (cot(4 - pi)))`
`= sin(4 - pi) = - sin 4 gt 0 ("as " sin 4 lt 0)`
(3) `tan (cos^(-1) (cos (2 pi - 5)))`
`= tan (2pi - 5) = -tan 5 gt 0 ("as " tan 5 lt 0)`
(4) `cot(sin^(-1) (sin (pi -4)) = cot (pi -4) = - cot 4 lt 0`