Property VIII:

(ii) tan^-1 x – tan^-1 y = tan^-1 [(x – y)/(1 + xy)], x ≥ 0, y ≥ 0.

Note:

For x < 0 and y < 0 these identities can be used with the help of properties (III) i.e., change x and y to-x and –y which are positive.

Property IX:

(i) sin^-1 x + sin^-1 y = sin^{-1, x ≥ 0, y ≥ 0 and x2 + y2 ≤ 1}

= π – sin^-1, x ≥ 0, y ≥ 0 and x² + y² > 1

(ii) sin^-1 x - sin^-1 y = sin^-1, x ≥ 0, y ≥ 0

Note:

For x < 0 and y < 0 these identities can be used with the help of properties (III) i.e., change x and y to –x and –y which are positive.

Property X:

(i) cos^-1 x + cos^-1 y = cos^-1,  x ≥ 0, y ≥ 0

(ii) cos^-1 x - cos^-1 y = cos^-1, x ≥ 0, y ≥ 0 and x ≤ y

= -cos^-1 x ≥ 0, y ≥ 0 and x > y

Note:

For x < 0 and y < 0 these identities can be used with the help of properties (III) i.e., change x and y to –x and –y which are positive.