Problem 34 - Contemporary Precalculus

Question

State whether or not the equation is an identity. If it is an identity, prove it. $$\sec ^{2} x+\csc ^{2} x=\sec ^{2} x \csc ^{2} x$$

Step-by-Step Solution

Answer

Question: Is the given equation an identity, and if so, prove it: \(\sec ^{2} x+\csc ^{2} x=\sec ^{2} x \csc ^{2} x\) Answer: Yes, the given equation is an identity. The proof is as follows: 1. Write the given equation: \(\sec ^{2} x+\csc ^{2} x=\sec ^{2} x \csc ^{2} x\) 2. Rewrite the equation using reciprocal functions: \(\left(\frac{1}{\cos^2 x}\right) + \left(\frac{1}{\sin^2 x}\right) = \left(\frac{1}{\cos^2 x}\right)\left(\frac{1}{\sin^2 x}\right)\) 3. Rewrite the equation using a common denominator: \(\frac{\sin^2 x + \cos^2 x}{\sin^2 x \cos^2 x} = \frac{1}{\sin^2 x \cos^2 x}\) 4. Use the sine and cosine relationship: \(\frac{1}{\sin^2 x \cos^2 x} = \frac{1}{\sin^2 x \cos^2 x}\) 5. Verify the equality: \(\frac{1}{\sin^2 x \cos^2 x} = \frac{1}{\sin^2 x \cos^2 x}\) Both sides of the equation are equal for all valid values of x, so the equation is an identity.

Step 1: Write the equation given in the exercise

Let's first rewrite the equation that we are given: $$\sec ^{2} x+\csc ^{2} x=\sec ^{2} x \csc ^{2} x$$

Step 2: Rewrite the equation using the reciprocal functions

Now, we need to rewrite the equation in terms of sine and cosine. Recall that \(\sec x = \frac{1}{\cos x}\) and \(\csc x = \frac{1}{\sin x}\). Using these reciprocal relations, we can rewrite the equation as: $$\left(\frac{1}{\cos^2 x}\right) + \left(\frac{1}{\sin^2 x}\right) = \left(\frac{1}{\cos^2 x}\right)\left(\frac{1}{\sin^2 x}\right)$$

Step 3: Rewrite the equation using a common denominator

To rewrite the equation using a common denominator, we can multiply both the numerator and denominator of the first term by \(\sin^2x\) and the numerator and denominator of the second term by \(\cos^2x\) : $$\frac{\sin^2 x + \cos^2 x}{\sin^2 x \cos^2 x} = \frac{1}{\sin^2 x \cos^2 x}$$

Step 4: Use sine and cosine relationship

We know that the relationship between sine and cosine is given by \(\sin^2 x + \cos^2 x = 1\). Using this relationship, we can simplify the left side of the equation: $$\frac{1}{\sin^2 x \cos^2 x} = \frac{1}{\sin^2 x \cos^2 x}$$

Step 5: Verify the equality

Comparing both sides of the equation, we can see that they are equal: $$\frac{1}{\sin^2 x \cos^2 x} = \frac{1}{\sin^2 x \cos^2 x}$$ Since both sides of the equation are equal for all valid values of x, the equation is an identity, and we have proven it.

Key Concepts

Reciprocal Functions

In trigonometry, reciprocal functions help us understand the relationships between the six primary trigonometric functions. The three reciprocal identities are:

• The secant function is the reciprocal of the cosine function: \(\sec x = \frac{1}{\cos x}\).
• The cosecant function is the reciprocal of the sine function: \(\csc x = \frac{1}{\sin x}\).
• The cotangent function is the reciprocal of the tangent function: \(\cot x = \frac{1}{\tan x}\).

These identities offer a convenient way to express trigonometric equations using different functions. For instance, if we have an equation involving \(\sec x\) or \(\csc x\), we can instead express it in terms of \(\cos x\) or \(\sin x\), respectively.
This method often simplifies calculations, as seen in our exercise, where expressing the original equation in terms of sine and cosine makes it easier to manipulate and verify.

Sine and Cosine Relationship

At the heart of trigonometry lies the fundamental relationship between sine and cosine. This is encapsulated in the Pythagorean identity:

• \(\sin^2 x + \cos^2 x = 1\)

This identity shows that the squares of sine and cosine of an angle add up to 1. It's central to many trigonometric proofs and equations. In our exercise, this relationship allows us to simplify the left side of the equation to 1 when combining the terms \(\sin^2 x\) and \(\cos^2 x\).
This identity results from the Pythagorean theorem, with the unit circle serving as a geometric visualization. The radius of the circle is 1, and for any angle \(x\), \(\cos x\) and \(\sin x\) can be viewed as the horizontal and vertical components, respectively.
Understanding this relationship is crucial when proving identities, as it often serves as the key simplifying step.

Common Denominator

When dealing with fractions in trigonometric identities, finding a common denominator is a technique that simplifies the equation. A common denominator creates a single expression from multiple terms, making it easier to manipulate mathematically.
Consider our exercise, where we rewrote the given equation:

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

We find the common denominator by multiplying each term appropriately:

• The common denominator becomes \(\sin^2 x \cos^2 x\).
• We adjust each fraction: multiply the first term by \(\frac{\sin^2 x}{\sin^2 x}\) and the second by \(\frac{\cos^2 x}{\cos^2 x}\).
• This results in: \(\frac{\sin^2 x + \cos^2 x}{\sin^2 x \cos^2 x}\)

Simplifying, and using the sine and cosine relationship \(\sin^2 x + \cos^2 x = 1\), reduces the equation on the left to \(\frac{1}{\sin^2 x \cos^2 x}\).
Applying this technique allows us to handle complex expressions and is a powerful tool for proving that an equation is an identity.