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Properties Of A Inverse Trigonometric Function

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Following are the properties of inverse trigonometric functions:
Inverse Trigonometric Functions

Proof of Property Number 1):
  • sin(sin-1x) = x

    Proof:

    Let (sin-1x) = y

    Hence sin(sin-1x) = sin y ⇒ x = sin y

    Hence sin(sin-1x) = x

    Similarly cos(cos-1x) = x, tan(tan-1x) = x, sec(sec-1x) = x, cosec(cosec-1x) = x and cot(cot -1x) = x can also be proved.

  • sin-1(sin x) = x

    Proof:

    Let sin x = y

    Hence sin-1(sin x) = sin-1y = 0

    But sin x = y ⇒ sin-1y = x

    Hence sin-1(sin x) = sin-1y = x

    Similarly other parts of b) can also be proved.

    Proof of Property Number 2):

    sin-1 = cosec-1x

    Proof:

    Let cosec-1x = y

    Hence x = cosec y ⇒ = sin y ⇒ sin-1 =y

    Hence, sin-1 = cosec-1x

    Similarly other parts can also be proved.

    Proof of Property Number 3):

    sin-1(-x) = -sin-1x

    Proof:

    Let sin-1(-x) = y = 0

    Hence -x = sin y

    Or         x = -sin y = sin(-y)

    Hence x = sin(-y) ⇒ sin-1x = -y or y = -sin-1x

    Similarly tan and cosec can also be proved.

    cos-1(-x) = π - cos-1x

    Proof:

    Let cos-1(-x) = y

    Hence -x = cos y or x = -cos y = cos(π - y)

    Hence cos-1x = π - y ⇒ y = π - cos-1x

    Hence, cos-1(-x) = π - cos-1x

    Similarly sec and cot can also be proved.

    Proof of Property Number 4):

    sin-1x + cos-1x =

    Proof:

    Let sin-1x = y

    Hence x = sin y = cos(- y)

    Hence cos-1x = - y or cos-1x- sin-1x

    Hence, sin-1x + cos-1x =

    Similarly the other 2 parts can also be proved.

    Proof of Property Number 5):

    sin-1x ± sin-1y = sin-1[x√(1 - y2) ± y√(1 - x2)]

    Proof:

    Let sin-1x = and sin-1y = β

    Hence x = sin and y = sinβ

    Then sin(± β) = sincosβ ± cos sinβ

    = sin√(1 - sin2β) ± sinβ√(1 - sin2)

    = x√(1 - y2) ± y√(1 - x2)

    Hence, ± β = sin-1[x√(1 - y2) ± y√(1 - x2)]

    Thus, sin-1x ± sin-1y = sin-1[x√(1 - y2) ± y√(1 - x2)]

    Similarly, the other two parts can also be proved.

    Proof of Property Number 6):


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