$cos^{-1}(cos(-5))+sin^{-1}(sin(6))-tan^{-1}(tan(12)) $ is equal to:

(The inverse trigonometric functions take the principal values)

(A) $3\pi – 11$
(B) $3\pi + 1 $
(C) $4\pi – 11$
(D) $4\pi – 9$

Solution

The given expression can be rewritten as,

$cos^{-1}cos5+sin^{-1}sin 6-tan^{-1} tan12 $

Considering the principal values the given expression can be further rewritten as,

$cos^{-1}cos(2\pi – 5)+sin^{-1}sin (-(2\pi- 6))-tan^{-1} tan(-(4\pi -12)) $

Or $(2\pi – 5)+ (-(2\pi- 6))-(-(4\pi-12)) = 4\pi -11 $

Answer: (C)