Problem 80 - Precalculus: Functions and Graphs

Question

Use the graph of \(f\) to find the simplest expression \(g(x)\) such that the equation \(f(x)=g(x)\) is an Identity. Verify this identity. $$f(x)=\frac{\sin ^{3} x+\sin x \cos ^{2} x}{\csc x}+\frac{\cos ^{3} x+\cos x \sin ^{2} x}{\sec x}$$

Step-by-Step Solution

Answer

The simplest expression is \( g(x) = 1 \), verified by simplifying to 1.

Step 1: Simplify the First Fraction

To simplify \( f(x) = \frac{\sin^3 x + \sin x \cos^2 x}{\csc x} \), we recognize that \( \csc x = \frac{1}{\sin x}\). Thus, \( \frac{\sin^3 x + \sin x \cos^2 x}{\csc x} = (\sin^3 x + \sin x \cos^2 x) \times \sin x = \sin^4 x + \sin^2 x \cos^2 x \).

Step 2: Simplify the Second Fraction

To simplify \( \frac{\cos^3 x + \cos x \sin^2 x}{\sec x} \), we use the identity \( \sec x = \frac{1}{\cos x}\). Thus, \( \frac{\cos^3 x + \cos x \sin^2 x}{\sec x} = (\cos^3 x + \cos x \sin^2 x) \times \cos x = \cos^4 x + \cos^2 x \sin^2 x \).

Step 3: Combine and Simplify the Expression

Combine the results from Steps 1 and 2: \( \sin^4 x + \sin^2 x \cos^2 x + \cos^4 x + \cos^2 x \sin^2 x \). Simplifying this, we use the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \), resulting in the expression: \( \sin^4 x + 2\sin^2 x \cos^2 x + \cos^4 x = (\sin^2 x + \cos^2 x)^2 = 1^2 = 1 \).

Step 4: Verify the Identity

Verify by confirming that each part simplifies correctly. The entire expression simplifies to \( 1 \), confirming that \( f(x) = 1 \). Therefore, \( g(x) = 1 \) is the simplest expression such that \( f(x) = g(x) \) is an identity.

Key Concepts

Simplification

Simplification is the process of reducing an expression to its simplest form without changing its value. This is a crucial skill in mathematics, especially when working with complex trigonometric expressions like the one given in the original problem. Simplification often involves several steps and makes use of algebraic and trigonometric identities.

In our exercise, the initial expression denominates with reciprocal trigonometric functions such as cosecant (\( \csc x \), which equals \( \frac{1}{\sin x} \)) and secant (\( \sec x \), which equals \( \frac{1}{\cos x} \)). Recognizing these relationships allows us to eliminate the denominators by multiplying:

• \( \frac{\sin^3 x + \sin x \cos^2 x}{\csc x} \) becomes \( (\sin^3 x + \sin x \cos^2 x) \times \sin x = \sin^4 x + \sin^2 x \cos^2 x \).
• \( \frac{\cos^3 x + \cos x \sin^2 x}{\sec x} \) becomes \( (\cos^3 x + \cos x \sin^2 x) \times \cos x = \cos^4 x + \cos^2 x \sin^2 x \).

Combining these simplified parts results in a single, more manageable expression. This simplifies the process of verifying whether the expression constitutes an identity.

Pythagorean Identity

The Pythagorean identity is one of the fundamental identities in trigonometry. It states that for any angle \(x\), the square of sine plus the square of cosine equals one: \[\sin^2 x + \cos^2 x = 1\]This identity is powerful and often simplifies complex trigonometric expressions efficiently.

In the given exercise, after simplifying the individual fractions, we end up with the expression: \[\sin^4 x + 2\sin^2 x \cos^2 x + \cos^4 x\]This can be recognized as the square of something familiar if you know your identities. By substituting \( \sin^2 x + \cos^2 x = 1 \) into the equation above, our expression corresponds to the known identity:\[(\sin^2 x + \cos^2 x)^2 = 1^2 = 1\]This demonstrates how the Pythagorean identity can be applied to simplify and validate trigonometric expressions. It provides a pathway to potentially revealing the identity of a more complicated function, confirming it sums to unity.

Identity Verification

Identity verification in mathematics involves proving that an equation holds true for all values of the variable it encompasses. This means the expression on the left-hand side and the right-hand side yield the same result, showing they are identically equal.

In solving the exercise, once we simplified the given function \(f(x)\), we ended up with the expression equivalent to \(1\). By applying trigonometric identities and simplifying systematically, we verified that:\[f(x) = \sin^4 x + 2\sin^2 x \cos^2 x + \cos^4 x = 1\]Thus, the most straightforward expression \(g(x)\) that shows \(f(x) = g(x)\) becomes \(g(x) = 1\). This verifies that the equation \(f(x) = g(x)\) is indeed an identity. It's a key step because it confirms a profound mathematical truth, ensuring the expression maintains its value across all its domain.