Property I:
(i)
sin-¹ (sin θ) = θ for
all θ ϵ [-π / 2, π / 2]
(ii)
cos-¹ (cos θ) = θ for
all θ ϵ [0, π]
(iii)
tan-¹ (tan θ) = θ for
all θ ϵ [-π / 2, π / 2]
(iv)
cosec-¹ (cosec θ) = θ for all θ ϵ [-π / 2, π / 2]
(v)
sec-¹ (sec θ) = θ for all θ ϵ [0, π], θ ≠ π / 2
(vi)
cot-¹ (cot θ) = θ for alll θ ϵ [0, π]
Property II:
(i)
sin (sin-¹ x) = x for
all x ϵ [-1, 1]
(ii)
cos (cos-¹ x) = x for
all x ϵ [-1, 1]
(iii)
tan (tan-¹ x) = x for
all x ϵ R
(iv)
cosec (cosec-¹ x) = x for all x ϵ [-∞, -1] U [1, ∞]
(v)
sec (sec-¹ x) = x for all x ϵ [-∞, -1] U [1, ∞]
(vi)
cot (cot-¹ x) = x for all x ϵ R
REMARK: It
should be noted that sin-¹ (sin θ) ≠ θ, if all θ ∉ [-π / 2, π / 2].
In fact, we have.
Similarly,
we have
Property III:
(i)
sin-¹ (-x) = - sin-¹ x, for all x ϵ [-1, 1]
(ii)
cos-¹ (-x) = π - cos-¹ x, for all x ϵ [-1, 1]
(iii)
tan-¹ (-x) = - tan-¹ x, for all x ϵ R
(iv)
cosec-¹ (-x) = - cosec-¹ x, for all x ϵ [-∞, -1] U [1, ∞]
(v)
sec-¹ (-x) = π - sec-¹ x, for all x ϵ [-∞, -1] U [1, ∞]
(vi)
cot-¹ (-x) = π - cot-¹ x, for all x ϵ R.
Property IV:
(i)
sin-¹ (1/x) = cosec-¹ x, for all x ϵ [-∞, -1] U [1, ∞]
(ii)
cos-¹ (1/x) = sec-¹ x, for all x ϵ [-∞, -1] U [1,
∞]
(iii) 
Property V:
(i)
sin-¹ x + cos-¹ x = π/2, for all x ϵ [-1, 1]
(ii)
tan-¹ x + cot-¹ x = π/2, for all x ϵ R
(iii)
sec-¹ x + cosec-¹ x = π/2, for all x ϵ [-∞, -1] U [1, ∞]
Property VI:
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