Question

Prove the following identities. $$\cos ^{-1} x+\cos ^{-1}(-x)=\pi$$

Short answer

In this problem, we prove the trigonometric identity \(\cos^{-1}(x) + \cos^{-1}(-x) = \pi\). We begin by defining the angles \(\alpha = \cos^{-1}(x)\) and \(\beta = \cos^{-1}(-x)\) and, by definition, we have that \(x = \cos(\alpha)\) and \(-x = \cos(\beta)\). Next, we use the cosine difference formula to find the cosine of the sum of the two angles. After substituting the expressions for the angles, we simplify the expression and find that \(\cos(\alpha + \beta) = -1\), which implies that the sum of the two angles is equal to \(\alpha + \beta = \cos^{-1}(-1) = \pi\). Finally, we conclude that \(\cos^{-1}(x) + \cos^{-1}(-x) = \pi\), and the identity is proven.

Step-by-step solution

Understand inverse cosine function properties

By definition, the inverse cosine function, written as \(\cos^{-1}(x)\), returns an angle in the interval \([0, \pi]\) for which cosine function returns the given value. In other words, if \(\alpha = \cos^{-1}(x)\), then \(x = \cos(\alpha)\). The same applies for \(\cos^{-1}(-x)\).

Write down the angles

Let \(\alpha = \cos^{-1}(x)\) and \(\beta = \cos^{-1}(-x)\). Then, by definition, we have: $$ x = \cos(\alpha) \text{ and } -x = \cos(\beta) $$

Use the cosine difference formula

Now, let's use the cosine difference formula to find the cosine of the sum of the two angles: $$ \cos(\alpha + \beta) = \cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta) $$

Replace angles with their expressions

We know that \(x = \cos(\alpha)\) and \(-x = \cos(\beta)\). We can rewrite the sine terms using the Pythagorean identity: \(\sin^2(\theta) = 1 - \cos^2(\theta)\). We obtain: $$ \cos(\alpha + \beta) = x(-x) - \sqrt{1 - x^2}\sqrt{1 - (-x)^2} $$

Simplify the expression

The expression becomes: $$ \cos(\alpha + \beta) = -x^2 - (1 - x^2) = -1 $$

Find the sum of the two angles

Since \(\cos(\alpha + \beta) = -1\), we find that \(\alpha + \beta = \cos^{-1}(-1) = \pi\).

Conclude the proof

Finally, combining the two definitions \(\alpha = \cos^{-1}(x)\) and \(\beta = \cos^{-1}(-x)\) and the sum of the two angles, we have \(\cos^{-1}(x) + \cos^{-1}(-x) = \alpha + \beta = \pi\). Hence, the identity is proven.