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Magazine Chapter 7: Problem 44
State whether or not the equation is an identity. If it is an identity, prove it. $$\cot ^{2} x-\cos ^{2} x=\cos ^{2} x \cot ^{2} x$$
Short Answer
Expert verified
Answer: Yes, the given equation is an identity.
Step by step solution
01
Write down the given equation
We start by writing down the equation we want to investigate:
$$\cot ^{2} x-\cos ^{2} x=\cos ^{2} x \cot ^{2} x$$
02
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:
$$\frac{\cos^2x}{\sin^2x} - \cos^2x = \cos^2x\frac{\cos^2x}{\sin^2x}$$
03
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:
$$\frac{\cos^2x-\cos^2x \sin^2x}{\sin^2x} = \cos^2x\frac{\cos^2x}{\sin^2x}$$
04
Simplify the terms
Now, we can simplify the terms on both sides of the equation:
$$\frac{\cos^2x(1 - \sin^2x)}{\sin^2x} = \frac{\cos^4x}{\sin^2x}$$
05
Use the Pythagorean identity
We apply the Pythagorean identity, \(1-\sin^2x=\cos^2x\), which allows us to further simplify the equation:
$$\frac{\cos^2x\cos^2x}{\sin^2x} = \frac{\cos^4x}{\sin^2x}$$
06
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.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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 \( \sin{x} = 0 \).
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 \(2\pi\).
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.
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.
