Basic Concept
* The principal value branch of sin^{-1} is [ – \frac{π}{2} , \frac{π}{2} ].
* The principal value branch of cos^{-1} is [ 0 , π ].
* The principal value branch of cosec^{-1}is [ –\frac{π}{2} , \frac{π}{2} ] – { 0 }.
* The principal value branch of tan^{-1} is ( – sec^{-1} ).
* The principal value branch of sec^{-1} is [0, π ] – { \frac{π}{2} }.
* The principal value branch of cot^{-1} is (0, π ).
Properties of Inverse Trigonometric Functions
Property 1:
- sin^{-1} \frac{1}{x} = cosec^{-1} x, x ≥ 1 or ≤ – 1
- cos^{-1} \frac{1}{x} = sec^{-1} x, x ≥ 1 or x ≤ – 1
- tan^{-1} \frac{1}{x} = cot^{-1} x, x > 0
Property 2:
- sin^{-1} ( − x ) – sin^{-1} ( x), x ∈ [ – 1, 1 ]
- tan^{-1} ( − x) = – tan^{-1} ( x), x ∈ R
- cosec^{-1} ( − x ) = – cosec^{-1} ( x), \mid x \mid ≥ 1
Property 3:
- cos^{-1} ( − x ) = π – cos^{-1} x, x ∈ [ – 1, 1]
- sin^{-1} ( − x ) = π – sec^{-1} x, \mid x \mid ≥ 1
- cot^{-1} ( − x ) = π – cot^{-1} x, x ∈ R
Property 4:
- sin^{-1} ( x ) + cos^{-1} ( x ) = \frac{π}{2} , x ∈ [ – 1, 1 ]
- tan^{-1} ( x) + cot^{-1} ( x ) = \frac{π}{2} , x ∈ R
- cosec^{-1} ( x ) + sec^{-1} ( x ) = \frac{π}{2} , \mid x \mid ≥ 1
Property 5:
- tan^{-1} x + tan^{-1} y = tan − 1 \frac{( x + y)}{1 − xy )} , xy < 1
- tan^{-1} x + tan^{-1} y = tan − 1 \frac{( x − y)}{1 + xy )} , xy < – 1
Property 6:
- 2tan^{-1} x = sec^{-1} \frac{( 2x)}{1 + x² )}, \mid x \mid ≤ 1
- 2tan^{-1} x = cos^{-1} \frac{( 1 − x²)}{1 + x² )} , x ≥ 0
- 2tan^{-1} x = tan^{-1} \frac{( 2x)}{1 − x² )} , – 1 < x < 1