Problem 64 - Precalculus: Functions and Graphs

Question

Either show that the equation \(i s\) an identity or show that the equation \(is\quad not\) an identity. $$\csc ^{2} x+\sec ^{2} x=\csc ^{2} x \sec ^{2} x$$

Step-by-Step Solution

Answer

The equation is an identity.

Step 1: Understand the Problem

We need to determine whether the trigonometric equation \( \csc ^{2} x+\sec ^{2} x=\csc ^{2} x \sec ^{2} x \) holds true for all allowed values of \( x \). If it holds for every valid input, it's an identity; otherwise, it's not.

Step 2: Recall Trigonometric Identities

Recall the definitions:\( \csc x = \frac{1}{\sin x} \) and \( \sec x = \frac{1}{\cos x} \). Therefore, \( \csc^2 x = \frac{1}{\sin^2 x} \) and \( \sec^2 x = \frac{1}{\cos^2 x} \). Substitute these into the given equation.

Step 3: Substitute into the Equation

Substitute \( \csc^2 x = \frac{1}{\sin^2 x} \) and \( \sec^2 x = \frac{1}{\cos^2 x} \) into the equation:\[ \frac{1}{\sin^2 x} + \frac{1}{\cos^2 x} = \frac{1}{\sin^2 x} \cdot \frac{1}{\cos^2 x} \].

Step 4: Find a Common Denominator

Combine the left side into a single fraction:\[ \frac{1}{\sin^2 x} + \frac{1}{\cos^2 x} = \frac{\cos^2 x + \sin^2 x}{\sin^2 x \cos^2 x} \].

Step 5: Simplify Using Pythagorean Identity

Recognize that \( \sin^2 x + \cos^2 x = 1 \), then substitute:\[ \frac{\cos^2 x + \sin^2 x}{\sin^2 x \cos^2 x} = \frac{1}{\sin^2 x \cos^2 x} \].

Step 6: Reduce the Right Hand Side

Recall that the right side of the original equation is:\[ \frac{1}{\sin^2 x \cos^2 x} \], which matches the simplified left side.

Step 7: Conclusion

Since both sides of the equation are equal, \( \frac{1}{\sin^2 x \cos^2 x} = \frac{1}{\sin^2 x \cos^2 x} \), the given equation is an identity.

Key Concepts

Cosecant Function

The cosecant function, denoted as \( \csc x \), is a fundamental trigonometric function closely related to the sine function. It is defined as the reciprocal of sine:

• \( \csc x = \frac{1}{\sin x} \)

This definition means that wherever \( \sin x \) is zero, \( \csc x \) is undefined—typically at integer multiples of \( \pi \).
To express \( \csc^2 x \), we square the basic definition of the cosecant, resulting in:

• \( \csc^2 x = \left( \frac{1}{\sin x} \right)^2 = \frac{1}{\sin^2 x} \)

In trigonometric identities and equations, \( \csc^2 x \) often shows up in collaboration with other similar functions like secant, especially when dealing with reciprocal relations or identities.

Secant Function

The secant function, denoted as \( \sec x \), is another key trigonometric function defined as the reciprocal of the cosine function:

• \( \sec x = \frac{1}{\cos x} \)

Just like the cosecant, the secant function is undefined where its base function, cosine, equals zero—this typically occurs at odd multiples of \( \frac{\pi}{2} \).
When squaring the secant function, we get:

• \( \sec^2 x = \left( \frac{1}{\cos x} \right)^2 = \frac{1}{\cos^2 x} \)

This squared secant function forms part of various trigonometric identities and plays a crucial role in understanding complex trigonometric equations, including those involving derivatives or integrals.

Pythagorean Identity

The Pythagorean identity is a cornerstone of trigonometry. It's derived from the Pythagorean theorem and relates the squares of the sine and cosine functions:

• \( \sin^2 x + \cos^2 x = 1 \)

This identity is essential because it serves as a foundation for manipulating and simplifying trigonometric expressions.
For example, in the given problem, understanding that \( \sin^2 x + \cos^2 x = 1 \) allows us to replace these sums in the equation:

• \( \frac{\cos^2 x + \sin^2 x}{\sin^2 x \cos^2 x} = \frac{1}{\sin^2 x \cos^2 x} \)

This simplification matches both sides of the equation, demonstrating its identity status.
By grasping the Pythagorean identity, students can tackle a wide range of trigonometric problems with more confidence, knowing that the sum of squares of sine and cosine will always equal one.