Problem 47 - Algebra and Trigonometry

Question

Rewrite the expression as an algebraic expression in \(x .\) \(\cos \left(\cos ^{-1} x+\sin ^{-1} x\right)\)

Step-by-Step Solution

Answer

The expression simplifies to 0.

Step 1: Understand the Expression

We need to rewrite the expression \(\cos \left(\cos^{-1} x + \sin^{-1} x\right)\) using an algebraic form in terms of \(x\). This involves trigonometric identities.

Step 2: Recall Trigonometric Identities

The inverse trigonometric function properties state that \(\cos(\cos^{-1}(x)) = x\) and \(\sin(\sin^{-1}(x)) = x\). It is also important to remember that \(\cos^{-1}(x) + \sin^{-1}(x) = \frac{\pi}{2}\) for \(-1 \leq x \leq 1\).

Step 3: Express Using Identity

Using the identity from Step 2, substitute to get \(\cos\left(\frac{\pi}{2}\right)\) because \(\cos^{-1} x + \sin^{-1} x = \frac{\pi}{2}\).

Step 4: Simplify the Expression

Since \(\cos\left(\frac{\pi}{2}\right) = 0\), the expression simplifies to 0. Therefore, the algebraic expression in \(x\) is 0.

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental tools in trigonometry, helping simplify complex expressions involving angles and their trigonometric functions. In this context, they are particularly useful for simplifying expressions that include inverse functions like \(\cos^{-1}(x)\) and \(\sin^{-1}(x)\).
Here, we utilized an important identity: \(\cos^{-1}(x) + \sin^{-1}(x) = \frac{\pi}{2}\), which holds for \(-1 \leq x \leq 1\). This identity tells us that the sum of the inverse cosine and inverse sine of the same angle equals \(\frac{\pi}{2}\).
Such identities are derived from the properties of right-angled triangles and the unit circle, making them invaluable for converting trigonometric expressions into algebraic forms.

Inverse Trigonometric Functions

Inverse trigonometric functions, like \(\cos^{-1}(x)\) and \(\sin^{-1}(x)\), are the "reverse" operations of the standard trigonometric functions. They help us find angles when we know the side ratios. For example, \(\cos^{-1}(x)\) returns the angle whose cosine value is \(x\).
It's crucial to understand the domain and range of these functions to use them properly. The range of \(\cos^{-1}(x)\) is \([0, \pi]\), while for \(\sin^{-1}(x)\), it is \([-\frac{\pi}{2}, \frac{\pi}{2}]\). These ranges ensure that each function has one unique angle for every value of \(x\) in its domain.
These functions are key in transforming complex trigonometric expressions into simpler algebraic forms, making them more manageable to work with in various mathematical scenarios.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions involves reducing them to a more manageable or recognizable form, often by using identities or substitutions. In our exercise, simplification was achieved by substituting summed inverse trigonometric functions with a known identity.
This made it possible to convert \(\cos(\cos^{-1}(x) + \sin^{-1}(x))\) directly into 0 since \(\cos(\frac{\pi}{2}) = 0\).
Key steps in simplifying include identifying relevant identities, performing substitutions, and calculating any resulting standard trigonometric values. This not only makes the problem more straightforward but also helps in understanding the relationships between different trigonometric elements.