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Chapter 5: Problem 42

Simplify the expression. $$\frac{\cot ^{2} \alpha-4}{\cot ^{2} \alpha-\cot \alpha-6}$$

Short Answer

Expert verified
The simplified expression is \( \frac{\cot \alpha - 2}{\cot \alpha - 3} \).

Step by step solution

01

Factor the Numerator

The numerator of the expression is \( \cot^2 \alpha - 4 \). This can be viewed as a difference of squares: \( (\cot \alpha)^2 - 2^2 \). This can be factored into \((\cot \alpha + 2)(\cot \alpha - 2)\).
02

Factor the Denominator

The denominator is \( \cot^2 \alpha - \cot \alpha - 6 \). We need to find two numbers that multiply to \(-6\) and add to \(-1\). These numbers are \(-3\) and \(2\). Therefore, the denominator can be factored as \((\cot \alpha - 3)(\cot \alpha + 2)\).
03

Cancel Common Factors

The factored expression is now \( \frac{(\cot \alpha + 2)(\cot \alpha - 2)}{(\cot \alpha - 3)(\cot \alpha + 2)} \). The term \( \cot \alpha + 2 \) is common in both the numerator and the denominator, so they can be cancelled out.
04

Write the Simplified Expression

After cancelling the common factor, we are left with \( \frac{\cot \alpha - 2}{\cot \alpha - 3} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simplifying Expressions
Simplifying expressions involves reducing the expression to its most basic form. This is often done by canceling common factors in a fraction's numerator and denominator.

In our exercise, we started with the expression:
  • Numerator: \( \cot^2 \alpha - 4 \)
  • Denominator: \( \cot^2 \alpha - \cot \alpha - 6 \)
By factoring both the numerator and the denominator, we were able to identify common terms, which could then be canceled out.

Doing this simplification helps make the expression easier to work with. This is especially helpful for further calculations or when you need to plug in values to evaluate the expression.
Factoring Polynomials
Factoring polynomials is a method used to rewrite a polynomial as a product of simpler polynomials. It makes simplifying algebraic expressions easier.

For instance, consider the numerator in the expression, \( \cot^2 \alpha - 4 \). Since it is a difference of squares, it can be rewritten as:
  • \( (\cot \alpha)^2 - 2^2 \)
The numerator factors into \((\cot \alpha + 2)(\cot \alpha - 2)\).

The denominator is a trinomial, \( \cot^2 \alpha - \cot \alpha - 6 \). We found two numbers, -3 and 2, which add up to -1 and multiply to -6, allowing us to factor it as \((\cot \alpha - 3)(\cot \alpha + 2)\).

Knowing how to factor polynomials helps greatly in solving and simplifying expressions.
Trigonometry
Trigonometry deals with the relationships between angles and sides of triangles and the functions that describe these relationships. A key element in trigonometry is remembering the identities, such as cotangent (\( \cot \)) which is the reciprocal of the tangent function.

In the given expression, \( \cot \alpha \) represents the cotangent of angle \( \alpha \), which plays a vital role. Understanding these trigonometric functions allows us to manipulate and simplify expressions involving trig terms. For example:
  • \( \cot \alpha = \frac{1}{\tan\alpha} \)
  • Use of cotangent to factor expressions like \( (\cot \alpha + 2)(\cot \alpha - 2) \)
Grasping these fundamental trigonometric concepts makes it much easier to manage more complex algebraic and trigonometric problems.

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Most popular questions from this chapter

A point \(P\) in simple harmonic motion has a frequency of \(\frac{1}{2}\) oscillation per minute and an amplitude of 4 feet. Express the motion of \(P\) by means of an equation of the form \(d=a \sin \omega t\)

As \(x \rightarrow 0^{+}, f(x) \rightarrow L\) for some real number \(L\) Use a graph to predict \(L\) $$f(x)=x \cot x$$

Verify the identity by transforming the lefthand side into the right-hand side. $$\sec ^{2} 3 \theta \csc ^{2} 3 \theta=\sec ^{2} 3 \theta+\csc ^{2} 3 \theta$

: Rewrite the expression in nonradical form without using absolute values for the indicated values of $${1+\cot ^{2} \theta} ; \quad 0<\theta<\pi$$

: Rewrite the expression in nonradical form without using absolute values for the indicated values of $$\sqrt{\csc ^{2} \theta-1} ; \quad 3 \pi / 2<\theta<2 \pi$$
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