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$cos^{-1}cos(-5)+$

$sin^{-1}sin6$-

$tan^{-1}tan12 =? $

$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)