cos ⁻¹(cos ( - 5)) + sin ⁻¹(sin (6)) - tan ⁻¹(tan (12)) is equal to :
(The inverse trigonometric functions take the principal values)

Solution

Maths sir · 3 steps

Shift each angle by multiples of π or 2π until it lands in that function's principal range.

1. The rule. cos⁻¹(cos x), sin⁻¹(sin x) and tan⁻¹(tan x) give back x only when x is already in the principal range. Otherwise, find the angle in that range with the same cos, sin or tan value. Use π ≈ 3.14 to place each number.

   Term	Principal range	Angle that fits	Check
   cos⁻¹(cos(-5))	[0, π]	cos(-5) = cos 5 = cos(2π - 5), so 2π - 5	2π - 5 ≈ 1.28
   sin⁻¹(sin 6)	[-π/2, π/2]	sin 6 = sin(6 - 2π), so 6 - 2π	6 - 2π ≈ -0.28
   tan⁻¹(tan 12)	(-π/2, π/2)	tan 12 = tan(12 - 4π), so 12 - 4π	12 - 4π ≈ -0.57

2. Combine. Add the first two values and subtract the third.

   (2π - 5) + (6 - 2π) - (12 - 4π) = 4π - 11

3. So. The expression equals 4π - 11, which is option (c).

The idea to keep. Cosine repeats with period 2π and is even, sine repeats with period 2π, and tangent repeats with period π. Subtract the right multiple, then check the result is inside the principal range.

Answer from the NTA final answer key. Solution by PrepWiser.