Problem 44 - Contemporary Precalculus

Question

State whether or not the equation is an identity. If it is an identity, prove it. $$\cot ^{2} x-\cos ^{2} x=\cos ^{2} x \cot ^{2} x$$

Step-by-Step Solution

Answer

Answer: Yes, the given equation is an identity.

Step 1: Write down the given equation

We start by writing down the equation we want to investigate: $$\cot ^{2} x-\cos ^{2} x=\cos ^{2} x \cot ^{2} x$$

Step 2: Convert cotangent to cosine and sine

Knowing that \(\cot{x} = \frac{\cos{x}}{\sin{x}}\), we can rewrite the equation in terms of cosine and sine: $$\frac{\cos^2x}{\sin^2x} - \cos^2x = \cos^2x\frac{\cos^2x}{\sin^2x}$$

Step 3: Find a common denominator

In order to manipulate the left side of the equation, we look for a common denominator. Since both fractions have a denominator of \(\sin^2x\), we find that our common denominator is \(\sin^2x\). Then we have: $$\frac{\cos^2x-\cos^2x \sin^2x}{\sin^2x} = \cos^2x\frac{\cos^2x}{\sin^2x}$$

Step 4: Simplify the terms

Now, we can simplify the terms on both sides of the equation: $$\frac{\cos^2x(1 - \sin^2x)}{\sin^2x} = \frac{\cos^4x}{\sin^2x}$$

Step 5: Use the Pythagorean identity

We apply the Pythagorean identity, \(1-\sin^2x=\cos^2x\), which allows us to further simplify the equation: $$\frac{\cos^2x\cos^2x}{\sin^2x} = \frac{\cos^4x}{\sin^2x}$$

Step 6: Verify the equation is an identity

Now we observe that both sides of the equation are exactly the same: $$\frac{\cos^4x}{\sin^2x} = \frac{\cos^4x}{\sin^2x}$$ Since the equation holds true for all possible values of x (keeping in mind that the sine function cannot be zero at the same time), it is proven to be an identity.

Key Concepts

Cotangent Function

The cotangent function, denoted as \( \cot{x} \), is an important trigonometric ratio that relates the cosine and sine functions. It is defined as the reciprocal of the tangent function.

• \( \cot{x} = \frac{1}{\tan{x}} = \frac{\cos{x}}{\sin{x}} \)
• It represents the ratio of the adjacent side to the opposite side in a right triangle.
• Cotangent is periodic, repeating every \( \pi \), and is undefined whenever \( \sin{x} = 0 \).

Understanding cotangent is crucial when manipulating trigonometric expressions. In our exercise, spotting \( \cot{x} \) made us express it in terms of sine and cosine, which led to a simple equation that was easier to handle using identities.

Cosine Function

The cosine function, symbolized as \( \cos{x} \), is fundamental in trigonometry. It arises frequently in the analysis of right triangles and periodic functions.

• Defines the ratio of the adjacent side to the hypotenuse in a right triangle.
• Like sine, cosine ranges from -1 to 1 and is periodic, repeating every \(2\pi\).

In the equation from the exercise, \( \cos^2{x} \) appeared multiple times. Knowing properties of cosine helps simplify these terms. In particular, using the identity \( \cos^2{x} = 1 - \sin^2{x} \) allowed us to equate and simplify the expressions more effortlessly. By expressing cotangent in sine and cosine, we take advantage of this connection, easing the manipulation of trigonometric identities.

Pythagorean Identity

The Pythagorean Identity is one of trigonometry's key identities and it plays a pivotal role in simplifying trigonometric expressions.

• The basic identity is \( \sin^2{x} + \cos^2{x} = 1 \).
• It is derived from the Pythagorean theorem, which applies to right triangles.

In this exercise, the variation \( 1 - \sin^2{x} = \cos^2{x} \) was used. This allowed us to rewrite part of the equation as a \( \cos^2{x} \) expression, crucially simplifying the equation to verify that it equals the originally given expression on both sides.

Simplifying Equations

Simplifying equations, especially trigonometric ones, involves multiple steps and applications of identities. The process includes:

• Finding a common denominator: This step was used to combine terms into a single fraction.
• Substituting identities: Using known identities like \( \sin^2{x} + \cos^2{x} = 1 \) to replace terms.
• Simplifying expressions: Breaking down complex terms into simpler, equal forms.

By transforming all terms into sine and cosine (where possible), the exercise showed how two seemingly different mathematical expressions can be proven equivalent through identities. Each simplification step brings us closer to affirming (or refuting) that an equation is indeed an identity, and demonstrates the power of algebraic manipulation alongside trigonometric identities.