Inverse Trigonometry Sheet 4 (Revised)

Complementary Identities & Addition Formulas

1. Complementary Angle Identities

Identity Domain Conditions
$$\sin^{-1} x + \cos^{-1} x = \frac{\pi}{2}$$ $$x \in [-1, 1]$$
$$\tan^{-1} x + \cot^{-1} x = \frac{\pi}{2}$$ $$x \in \mathbb{R}$$
$$\sec^{-1} x + \csc^{-1} x = \frac{\pi}{2}$$ $$|x| \geq 1$$

2. Addition and Subtraction of Inverse Tangents

Formula Result Condition
$$\tan^{-1} x + \tan^{-1} y = \tan^{-1}\left(\frac{x+y}{1-xy}\right)$$ $$xy < 1$$
$$\tan^{-1} x + \tan^{-1} y = \pi + \tan^{-1}\left(\frac{x+y}{1-xy}\right)$$ $$xy > 1, x>0, y>0$$
$$\tan^{-1} x + \tan^{-1} y = -\pi + \tan^{-1}\left(\frac{x+y}{1-xy}\right)$$ $$xy > 1, x<0, y<0$$
$$\tan^{-1} x – \tan^{-1} y = \tan^{-1}\left(\frac{x-y}{1+xy}\right)$$ $$xy > -1$$

3. Conversion of $$2\tan^{-1} x$$

$$2\tan^{-1} x = \sin^{-1}\left(\frac{2x}{1+x^2}\right)$$ $$|x| \leq 1$$
$$2\tan^{-1} x = \cos^{-1}\left(\frac{1-x^2}{1+x^2}\right)$$ $$x \geq 0$$
$$2\tan^{-1} x = \tan^{-1}\left(\frac{2x}{1-x^2}\right)$$ $$-1 < x < 1$$
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