91Ó°ÊÓ

In trigonometry, identities are equations that hold true for any value in their domains. They are fundamental tools for simplifying complex expressions and solving equations. In the provided exercise, we use the well-known identity \(\cos^{-1} x + \sin^{-1} x = \frac{\pi}{2}\). This tells us that the sum of the arccosine and arcsine of a number equals \(\frac{\pi}{2}\).- This identity helps in transforming the original equation into a more manageable form.

By applying this identity, the expression \(\cos^{-1} x - \sin^{-1} x\) was rewritten as \(\frac{\pi}{2} - \sin^{-1} x = \cos^{-1} (x\sqrt{3})\). This shows how identities serve as powerful tools for manipulation and simplification in trigonometry.

Solving trigonometric equations often involves transforming the problem using identities and algebraic manipulation. In this exercise, we start by rearranging the given equation based on the identity \(\cos^{-1} x + \sin^{-1} x = \frac{\pi}{2}\). This simplifies the expression:

• Original: \(\cos^{-1} x - \sin^{-1} x = \cos^{-1}(x \sqrt{3})\)
• Rewritten: \(\frac{\pi}{2} = \cos^{-1}(x\sqrt{3}) + \sin^{-1} x\)

Next, applying the identity again, simplifies the problem further, leading to the equation \(x\sqrt{3} = x\). This is solved by examining each possibility:

• \(x = 0\) works because \(0 \times \sqrt{3} = 0\).
• \(\sqrt{3} = 1\) is impossible, leaving \(x = 0\) as the only solution.

This showcases how careful manipulation and knowledge of identities can streamline the problem-solving process.

The arccosine and arcsine are inverse functions of cosine and sine, respectively. Understanding their relationships helps in solving equations involving these functions. The important thing to note here is how these values interact, specifically with their domains:

• \(\cos^{-1} x\) results in values between \(0\) and \(\pi\).
• \(\sin^{-1} x\) results in values between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\).

An essential relationship between them is \(\cos^{-1} x + \sin^{-1} x = \frac{\pi}{2}\). This identity plays a critical role in this exercise, helping to rewrite and transform the initial equation for clarity. By understanding these inverse relationships, solving equations that involve arccosine and arcsine can become more intuitive, as these identities simplify complex trigonometric scenarios.