Important Results of Inverse Trigonometry

This lesson collects the standard inverse trigonometric identities used throughout Class 12, stating each result with the condition on $x$ where it holds. Together they form the toolkit for simplifying and solving inverse trigonometry questions.

Negative argument results

For a negative input, the sine and tangent inverses simply pick up a minus sign, while the cosine, secant, and cotangent inverses reflect about $\pi$.

$\sin^{-1}(-x) = -\sin^{-1} x$

$\tan^{-1}(-x) = -\tan^{-1} x$

$\cos^{-1}(-x) = \pi - \cos^{-1} x$

$\sec^{-1}(-x) = \pi - \sec^{-1} x$

$\cot^{-1}(-x) = \pi - \cot^{-1} x$

Complementary pairs

Each inverse function and its co-function add to a right angle.

$\sin^{-1} x + \cos^{-1} x = \tfrac{\pi}{2}$

$\tan^{-1} x + \cot^{-1} x = \tfrac{\pi}{2}$

$\sec^{-1} x + \csc^{-1} x = \tfrac{\pi}{2}$

Sine and cosine of an inverse function

Applying sine or cosine to the opposite inverse function produces the same square root.

$\sin\left(\cos^{-1} x\right) = \sqrt{1 - x^2}$

$\cos\left(\sin^{-1} x\right) = \sqrt{1 - x^2}$

Twice an inverse tangent

The quantity $2\tan^{-1} x$ can be rewritten as an inverse sine, an inverse cosine, or an inverse tangent, each valid on its own range.

$2\tan^{-1} x = \sin^{-1}\frac{2x}{1 + x^2}, \quad |x| \le 1$

$2\tan^{-1} x = \cos^{-1}\frac{1 - x^2}{1 + x^2}, \quad x \ge 0$

$2\tan^{-1} x = \tan^{-1}\frac{2x}{1 - x^2}, \quad -1 < x < 1$

Sum and difference of inverse tangents

Two inverse tangents combine into a single one, with the sign in the denominator opposite to the sign of the combination.

$\tan^{-1} x + \tan^{-1} y = \tan^{-1}\frac{x + y}{1 - xy}, \quad xy < 1$

$\tan^{-1} x - \tan^{-1} y = \tan^{-1}\frac{x - y}{1 + xy}, \quad xy > -1$

Key takeaways

• Sine and tangent inverses are odd, so a negative input flips the sign; cosine, secant, and cotangent inverses turn it into $\pi$ minus the value.
• An inverse function and its co-function always add to $\tfrac{\pi}{2}$.
• $2\tan^{-1} x$ has three equivalent forms as an inverse sine, cosine, or tangent, each holding on its stated range, and two inverse tangents add or subtract using the $\tan^{-1}\frac{x \pm y}{1 \mp xy}$ formula.