/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 31 Establish each identity. $$(\s... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 31

Establish each identity. $$(\sec \theta+\tan \theta)(\sec \theta-\tan \theta)=1$$

Short Answer

Expert verified
(\sec \theta + \tan \theta)(\sec \theta - \tan \theta) = 1 by recognizing it as a difference of squares and using the Pythagorean identity.

Step by step solution

01

Recognize the Difference of Squares

Notice that the left-hand side of the identity \[\begin{equation} (\sec \theta + \tan \theta)(\sec \theta - \tan \theta) \end{equation}\] is in the form of \[\begin{equation} (a + b)(a - b) = a^2 - b^2.\end{equation}\]
02

Set up for Substitution

Set \[\begin{equation} a = \sec \theta \end{equation}\] and \[\begin{equation} b = \tan \theta. \end{equation}\] Substitute into the formula for the difference of squares \[\begin{equation} a^2 - b^2.\end{equation}\]
03

Express in Terms of \sec \theta \

Substitute \[\begin{equation} a \end{equation}\] and \[\begin{equation} b \end{equation}\] into the formula: \[\begin{equation} \sec^2 \theta - \tan^2 \theta.\end{equation}\]
04

Use Trigonometric Identity

Recall the Pythagorean identity for secant and tangent: \[\begin{equation} \sec^2 \theta = 1 + \tan^2 \theta. \end{equation}\]
05

Substitute and Simplify

Substitute the identity \[\begin{equation} \sec^2 \theta = 1 + \tan^2 \theta \end{equation}\] into the expression: \[\begin{equation} \sec^2 \theta - \tan^2 \theta = (1 + \tan^2 \theta) - \tan^2 \theta. \end{equation}\]
06

Final Simplification

\[\begin{equation} (1 + \tan^2 \theta) - \tan^2 \theta = 1 \end{equation}\] which simplifies to 1. Therefore, \[\begin{equation} (\sec \theta + \tan \theta)(\sec \theta - \tan \theta) = 1. \end{equation}\]

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Key Concepts

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

Pythagorean Identities
In trigonometry, Pythagorean identities are essential tools that help simplify expressions. They originate from the Pythagorean theorem in geometry.
The basic Pythagorean identity is: \[ \sin^2 \theta + \cos^2 \theta = 1 \] This equation forms the basis for the derived identities.
For example, by dividing each term by \(\cos^2 \theta\), we obtain another useful identity: \[ \tan^2 \theta + 1 = \sec^2 \theta \] This identity is directly used in the given exercise.
Understanding these identities makes it easier to work with trigonometric expressions, as shown in the step-by-step solution.
Difference of Squares
The difference of squares is a powerful algebraic identity: \[ (a + b)(a - b) = a^2 - b^2 \] In this formula, \(a\) and \(b\) can represent any expressions.
This identity helps transform products into differences, making simplification manageable.
In the exercise, we used it by setting \(a = \sec \theta\) and \(b = \tan \theta\), leading to the expression \(\sec^2 \theta - \tan^2 \theta\).
Recognizing and applying the difference of squares can save time and simplify complex expressions quickly.
Secant and Tangent Relationship
The relationship between secant and tangent is given by one of the Pythagorean identities: \[ \sec^2 \theta = 1 + \tan^2 \theta \] This expresses \(\sec^2 \theta\) as a sum involving \(\tan^2 \theta\).
This identity shows that as tangent increases, secant also increases, maintaining the relationship.
In our exercise, substituting this identity into \(\sec^2 \theta - \tan^2 \theta\) led directly to the simplification: \[ \sec^2 \theta = 1 + \tan^2 \theta \] Substituted back, this became: \[ 1 + \tan^2 \theta - \tan^2 \theta = 1 \] Understanding how secant and tangent interact is key to solving many trigonometric equations.
Simplifying Trigonometric Expressions
Simplifying trigonometric expressions is a fundamental skill in trigonometry. It involves transforming complex expressions into their simplest forms using identities and algebraic techniques. Here are some steps to follow:
  • Identify applicable trigonometric identities.
  • Use algebraic methods like factoring, expanding, or using the difference of squares.
  • Substitute identities where applicable, such as \(\sec^2 \theta - \tan^2 \theta\).
  • Simplify step-by-step, ensuring each part is correct.
In our case, by recognizing the difference of squares and applying the identity \(\sec^2 \theta = 1 + \tan^2 \theta\), we simplified the expression to 1.
Practicing these simplifications improves problem-solving skills and mathematical agility.

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Most popular questions from this chapter

Area under a Curve The area under the graph of \(y=\frac{1}{1+x^{2}}\) and above the \(x\) -axis between \(x=a\) and \(x=b\) is given by $$ \tan ^{-1} b-\tan ^{-1} a $$ (a) Find the exact area under the graph of \(y=\frac{1}{1+x^{2}}\) and above the \(x\) -axis between \(x=0\) and \(x=\sqrt{3}\). (b) Find the exact area under the graph of \(y=\frac{1}{1+x^{2}}\) and above the \(x\) -axis between \(x=-\frac{\sqrt{3}}{3}\) and \(x=1\)

Movie Theater Screens Suppose that a movie theater has a screen that is 28 feet tall. When you sit down, the bottom of the screen is 6 feet above your eye level. The angle formed by drawing a line from your eye to the bottom of the screen and another line from your eye to the top of the screen is called the viewing angle. In the figure, \(\theta\) is the viewing angle. Suppose that you sit \(x\) feet from the screen. The viewing angle \(\theta\) is given by the function $$ \theta(x)=\tan ^{-1}\left(\frac{34}{x}\right)-\tan ^{-1}\left(\frac{6}{x}\right) $$ (a) What is your viewing angle if you sit 10 feet from the screen? 15 feet? 20 feet? (b) If there are 5 feet between the screen and the first row of seats and there are 3 feet between each row and the row behind it, which row results in the largest viewing angle? (c) Using a graphing utility, graph $$ \theta(x)=\tan ^{-1}\left(\frac{34}{x}\right)-\tan ^{-1}\left(\frac{6}{x}\right) $$ What value of \(x\) results in the largest viewing angle?

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Determine the points of intersection of the graphs of \(f(x)=x^{2}+5 x+1\) and \(g(x)=-2 x^{2}-11 x-4\) by solving \(f(x)=g(x)\)

Solve each equation on the interval \(0 \leq \theta<2 \pi\). $$ \sin \theta-\cos \theta=-\sqrt{2} $$

Area of a Dodecagon Part I A regular dodecagon is a polygon with 12 sides of equal length. See the figure. (a) The area \(A\) of a regular dodecagon is given by the formula \(A=12 r^{2} \tan \frac{\pi}{12},\) where \(r\) is the apothem, which is a line segment from the center of the polygon that is perpendicular to a side. Find the exact area of a regular dodecagon whose apothem is 10 inches. (b) The area \(A\) of a regular dodecagon is also given by the formula \(A=3 a^{2} \cot \frac{\pi}{12},\) where \(a\) is the length of a side of the polygon. Find the exact area of a regular dodecagon if the length of a side is 15 centimeters.
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