Basic identities

A small set of identities lets you rewrite almost any inverse trigonometric expression. This subtopic catalogues the foundational identities , those that do not require addition formulas. They follow from the principal range conventions and from basic trigonometric identities like $\sin^2 \theta + \cos^2 \theta = 1$.

Complementary identities

The most important identity in the chapter:

$\sin^{-1} x + \cos^{-1} x = \frac{\pi}{2}, \quad x \in [-1, 1].$

Proof. Let $\theta = \sin^{-1} x$, so $\sin \theta = x$ and $\theta \in [-\pi/2, \pi/2]$. Then $\cos(\pi/2 - \theta) = \sin \theta = x$, with $\pi/2 - \theta \in [0, \pi]$. Hence $\cos^{-1} x = \pi/2 - \theta$, and addition gives $\pi/2$. $\blacksquare$

By the same argument:

$\tan^{-1} x + \cot^{-1} x = \frac{\pi}{2}, \quad x \in \mathbb{R}.$

$\sec^{-1} x + \csc^{-1} x = \frac{\pi}{2}, \quad |x| \ge 1.$

These three identities are used in nearly every problem of the chapter.

Reciprocal identities

For $|x| \ge 1$ we have $\csc^{-1} x = \sin^{-1}(1/x)$ and $\sec^{-1} x = \cos^{-1}(1/x)$.

Reason. If $\theta = \csc^{-1} x$ then $\csc \theta = x$, so $\sin \theta = 1/x$. Since $\theta \in [-\pi/2, \pi/2] \setminus \{0\}$, we have $\theta = \sin^{-1}(1/x)$.

For $x > 0$ we also have $\cot^{-1} x = \tan^{-1}(1/x)$, but this fails for $x < 0$ , there one must write $\cot^{-1} x = \pi + \tan^{-1}(1/x)$.

Negative argument identities

Cross identities

Often we need to convert one inverse function into another. For $x \in [0, 1]$:

$\sin^{-1} x = \cos^{-1} \sqrt{1 - x^2} = \tan^{-1} \frac{x}{\sqrt{1 - x^2}}.$

Reason. If $\sin \theta = x$ with $\theta \in [0, \pi/2]$, then $\cos \theta = \sqrt{1 - x^2}$, hence $\cos^{-1}(\sqrt{1 - x^2}) = \theta$. And $\tan \theta = x/\sqrt{1 - x^2}$.

For $x \ge 0$, $\tan^{-1} x = \sin^{-1}(x/\sqrt{1 + x^2}) = \cos^{-1}(1/\sqrt{1 + x^2})$.

These conversions are essential for combining inverse trig terms into a single one.

Worked examples

Example 1. Simplify $\sin^{-1} x + \cos^{-1} x$ for $x = 0.3$.

By the identity, the sum is $\pi/2$, regardless of the value of $x$ (as long as $|x| \le 1$).

Example 2. Simplify $\sin^{-1}(\tfrac{3}{5}) + \cos^{-1}(\tfrac{3}{5})$.

Same identity: answer $\pi/2$.

Example 3. Compute $\tan^{-1}(\tfrac{1}{3}) + \cot^{-1}(\tfrac{1}{3})$.

Answer $\pi/2$.

Example 4. Show $\sin^{-1}(\sin \tfrac{2\pi}{3}) = \pi/3$.

$\sin(2\pi/3) = \sin(\pi - 2\pi/3) = \sin(\pi/3) = \sqrt 3/2$. The principal value answer is $\sin^{-1}(\sqrt 3/2) = \pi/3$.

Example 5. Express $\sin^{-1}(\tfrac{3}{5})$ in terms of $\cos^{-1}$.

If $\sin \theta = 3/5$ with $\theta \in [0, \pi/2]$, then $\cos \theta = 4/5$. So $\sin^{-1}(3/5) = \cos^{-1}(4/5)$.

Example 6. Show $\cos^{-1}(-1/2) = \pi - \cos^{-1}(1/2) = \pi - \pi/3 = 2\pi/3$.

Direct: $\cos(2\pi/3) = -1/2$ and $2\pi/3 \in [0, \pi]$. So the answer is $2\pi/3$.

Try it yourself

1. Evaluate $\sin^{-1}(\sqrt 2/2) + \cos^{-1}(\sqrt 2/2)$.
2. Evaluate $\tan^{-1}(\sqrt 3) + \cot^{-1}(\sqrt 3)$.
3. Show $\sec^{-1}(2) + \csc^{-1}(2) = \pi/2$.
4. Express $\sin^{-1}(4/5)$ as $\cos^{-1}$ of something.
5. Express $\sin^{-1}(5/13)$ as $\tan^{-1}$ of something.
6. Simplify $\sin^{-1}(-1/2) + \cos^{-1}(-1/2)$.
7. Evaluate $\tan^{-1}(-1) + \cot^{-1}(-1)$ , careful with branches.
8. Show $\cos^{-1}(-x) = \pi - \cos^{-1} x$ by direct verification at $x = 1/2$.
9. Show $\sin^{-1}(-3/5) + \cos^{-1}(3/5) = \pi/2 - 2 \sin^{-1}(3/5)$.
10. Find the value of $\sin^{-1}(1) + \cos^{-1}(1)$.
11. Evaluate $\tan^{-1}(2) + \cot^{-1}(2)$.
12. Verify the identity $\sec^{-1} x = \cos^{-1}(1/x)$ at $x = 2$.
13. For $x > 0$ show $\cot^{-1} x = \tan^{-1}(1/x)$.
14. For $x < 0$ find a correct identity replacing $\cot^{-1} x = \tan^{-1}(1/x)$.

Pitfalls / Tricks

• The identity $\sin^{-1} x + \cos^{-1} x = \pi/2$ holds for all $x \in [-1, 1]$, including the boundary values.
• $\cot^{-1} x = \tan^{-1}(1/x)$ only for $x > 0$; for $x < 0$ add $\pi$.
• Always restrict $x$ to the natural domain before applying any identity. Asking for $\sin^{-1}(2)$ is meaningless in real analysis.
• For symbolic manipulations, $\sin^{-1}(\sin \theta) = \theta$ only when $\theta \in [-\pi/2, \pi/2]$.
• The identity $\cos^{-1}(-x) = \pi - \cos^{-1} x$ is asymmetric (compare to $\sin$): the cosine inverse is not an odd function.

Next we tackle the additive identities for sums of two inverse trig terms.