    # Properties Of A Inverse Trigonometric Function Following are the properties of 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): Online Properties of a Inverse Trigonometric Function Help:

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