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Chapter 1: Problem 96

Show that each of the following statements is an identity by transforming the left side of each one into the right side. $$ \cos \theta(\sec \theta-\cos \theta)=\sin ^{2} \theta $$

Short Answer

Expert verified
The identity holds: \( \cos \theta(\sec \theta-\cos \theta)=\sin ^{2} \theta \).

Step by step solution

01

Rewrite Secant Function

The secant function can be rewritten using the reciprocal identity as \( \sec \theta = \frac{1}{\cos \theta} \). Substitute this into the expression to get:\[ \cos \theta \left( \frac{1}{\cos \theta} - \cos \theta \right) \]
02

Simplify the Expression

Expand the expression by distributing \( \cos \theta \) into each term of the parentheses:\[ \cos \theta \cdot \frac{1}{\cos \theta} - \cos \theta \cdot \cos \theta \]
03

Simplify Each Term

The first term simplifies to 1 since \( \cos \theta \cdot \frac{1}{\cos \theta} = 1 \), and the second term becomes \( \cos^2 \theta \) because \( \cos \theta \cdot \cos \theta = \cos^2 \theta \). Now it becomes:\[ 1 - \cos^2 \theta \]
04

Use Pythagorean Identity

Recall the Pythagorean identity \( \sin^2 \theta = 1 - \cos^2 \theta \). Substitute this identity into our simplified expression:\[ 1 - \cos^2 \theta = \sin^2 \theta \]
05

Conclude the Solution

Thus, we have successfully transformed the left side of the given statement into the right side, verifying that it is indeed an identity:\[ \cos \theta(\sec \theta-\cos \theta)=\sin ^{2} \theta \]

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Key Concepts

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

Reciprocal Identity
Trigonometric identities are a crucial stepping stone in understanding how trigonometric functions interrelate. One of the most fundamental identities in trigonometry is the reciprocal identity. This identity expresses a trigonometric function as the reciprocal of another. In this case, the reciprocal identity we focus on is between secant and cosine.
  • The secant function, denoted as \( \sec \theta \), is the reciprocal of the cosine function \( \cos \theta \). This means that \( \sec \theta = \frac{1}{\cos \theta} \).
  • Using reciprocal identities like this allows us to transform expressions and solve trigonometric equations more easily.
  • By rewriting \( \sec \theta \) in terms of \( \cos \theta \), we can simplify complex expressions, like the one in our problem statement, \( \cos \theta (\sec \theta - \cos \theta) \).
This transformation is often the first step in simplifying a trigonometric equation, making subsequent mathematical manipulations much more straightforward.
Pythagorean Identity
The Pythagorean identity is a powerful tool in trigonometry that relates sine and cosine functions in a simple and elegant equation. This identity is based on the Pythagorean theorem and is foundational for simplifying expressions and proving identities.
  • The most common Pythagorean identity is \( \sin^2 \theta + \cos^2 \theta = 1 \).
  • By rearranging this identity, we can express \( \sin^2 \theta \) as \( 1 - \cos^2 \theta \), which plays a critical role in transforming trigonometric expressions.
  • In our problem, after simplifying with reciprocal identity, what remains is \( 1 - \cos^2 \theta \), which directly matches \( \sin^2 \theta \) thanks to the Pythagorean identity.
This identity allows us to recognize and transform portions of the given equation into more manageable or recognizable forms, eventually proving the original trigonometric identity.
Simplifying Expressions
Simplifying expressions is a central aspect of solving trigonometric identities and equations. This process involves rewriting a given expression in a form that is easier to work with or provides insight into the problem.
  • First, apply algebraic techniques, such as distributing terms, as done here with the expression \( \cos \theta (\sec \theta - \cos \theta) \).
  • Use trigonometric identities, like the reciprocal and Pythagorean identities, to alter the expression into a simpler or more meaningful form.
  • The primary goal in simplification is to reduce complexity, which can help in verifying identities, solving equations, or making connections with other mathematical principles.
In the example given, simplification started with substituting \( \sec \theta \) with \( \frac{1}{\cos \theta} \) and expanded further to \( 1 - \cos^2 \theta \), which, by using known identities, completes the proof of the identity.

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Most popular questions from this chapter

Find \(\sec \theta\) if \(\tan \theta=\frac{20}{21}\) and \(\cos \theta<0\).

Show that each of the following statements is an identity by transforming the left side of each one into the right side. $$ \sec \theta(\sin \theta+\cos \theta)=\tan \theta+1 $$

Use the equivalent forms of the first Pythagorean identity on Problems 31 through 38 . Find \(\tan \theta\) if \(\sin \theta=\frac{1}{3}\) and \(\theta\) terminates in QI.

Use the reciprocal identities for the following problems. If \(\sec \theta=-2\), find \(\cos \theta\).

Use the equivalent forms of the first Pythagorean identity on Problems 31 through 38 . Find \(\csc \theta\) if \(\cot \theta=\frac{24}{7}\) and \(\theta\) terminates in QIII.
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