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# Theory Part 2

**Property II: **

**(i) **sin (sin^{-1} x) = x= cos (cos^{-1} x), for all x ∈ [-1, 1]

**Property III: **

**(i) **sin^{-1} (-x) = -sin^{-1} (x), for all x ∈ [-1, 1]

**(ii) **cos^{-1} (-x) = π - cos^{-1} (x), for all x ∈ [-1, 1]

**(iii) **tan^{-1} (-x) = -tan^{-1} x for all x ∈ R

**(iv) **cosec^{-1} (-x) = -cosec^{-1} x, for all x ∈ [-∞, -1] υ [1, ∞)

**(v) **sec^{-1} (-x) = π - sec^{-1} x, for all x ∈ [-∞, -1] υ [1, ∞)

**(vi) **cot^{-1} (-x) = π - cot^{-1} x, for all x ∈ R

**Property IV:**

**(i) **sin^{-1} x + cos^{-1} x = (π/2), for all x ∈ [-1, 1]

**(ii) **tan^{-1} x + cot^{-1} x = (π/2), for all x ∈ R

**(iii) **sec^{-1} x + cosec^{-1} x = (π/2), for all x ∈ (-∞, -1] υ [1, ∞)

**Property V:**

**(i) **sin^{-1} (1/x) = cosec^{-1} x, for all x ∈ (-∞, -1] υ [1, ∞)

**(ii) **cos^{-1} (1/x) = sec^{-1} x, for all x ∈ (-∞, -1] υ [1, ∞)

**Property VI: **

**(i) **sin^{-1} x = cos^{-1} (1 – x^{2})^{1/2} = tan^{-1} [x/(1 – x^{2})^{1/2}]

= cot^{-1} [((1 – x^{2})^{1/2})/x] = sec^{-1} [1/(1 – x^{2})^{1/2}] = cosec^{-1} (1/x), x ∈ (0, 1)

**(ii) **cos^{-1} x = sin^{-1} (1 – x^{2})^{1/2} = tan^{-1} [((1 – x^{2})^{1/2})/x]

= cot^{-1} [x/(1 – x^{2})^{1/2}] = sec^{-1} (1/x) = cosec^{-1} [1/(1 – x^{2})^{1/2}], x ∈ (0, 1)

**(iii) **tan^{-1} x = sin^{-1} [x/(1 + x^{2})^{1/2}] = cos^{-1} [1/(1 + x^{2})^{1/2}]

= cot^{-1} (1/x) = sec^{-1} [(1 + x^{2})^{1/2}] = cosec^{-1} [((1 + x^{2})^{1/2})/x], x > 0

**Property VII: **

**(i) **sin (cos^{-1} x) = cos (sin^{-1} x) = (1 – x^{2})^{1/2}, -1 ≤ x ≤ 1.

**(ii) **tan (cot^{-1} x) = cot (tan^{-1} x) = (1/x), x ∈ R, x ≠ 0.

**(iii) **cosec (sec^{-1} x) = sec (cosec^{-1} x) =. |x| > 1.