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Chapter 1: Problem 31

Trigonometric identities $$\text { Prove that } \tan ^{2} \theta+1=\sec ^{2} \theta$$

Short Answer

Expert verified
Question: Prove the trigonometric identity \(\tan^2 \theta + 1 = \sec^2 \theta\). Answer: Following the steps outlined in the solution above, we were able to manipulate and simplify the left side of the equation, \(\tan^2 \theta + 1\), to arrive at the right side of the equation, \(\sec^2 \theta\). This demonstrates that the trigonometric identity \(\tan^2 \theta + 1 = \sec^2 \theta\) holds true.

Step by step solution

01

Write the left side in terms of sine and cosine functions

The left side of the equation is \(\tan^2 \theta + 1\). To begin, let's rewrite the tangent function in terms of the sine and cosine functions: \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). So, our left side becomes: $$\left(\frac{\sin \theta}{\cos \theta}\right)^2 + 1$$
02

Simplify the expression

Now, we can simplify the left side and write it as a single fraction: $$\frac{\sin^2 \theta}{\cos^2 \theta} + 1$$
03

Rewrite 1 as a fraction with the same denominator

To further simplify the expression, we need to rewrite the number 1 in terms of a fraction with the same denominator as the first term. We will rewrite 1 as \(\frac{\cos^2 \theta}{\cos^2 \theta}\): $$\frac{\sin^2 \theta}{\cos^2 \theta} + \frac{\cos^2 \theta}{\cos^2 \theta}$$
04

Combine the fractions

Now, add the two fractions together: $$\frac{\sin^2 \theta + \cos^2 \theta}{\cos^2 \theta}$$
05

Use the Pythagorean identity

Since we know that \(\sin^2 \theta + \cos^2 \theta = 1\), we can substitute 1 for the numerator of the fraction: $$\frac{1}{\cos^2 \theta}$$
06

Rewrite in terms of secant function

Finally, we can rewrite the expression in terms of the secant function, which is defined as \(\sec \theta = \frac{1}{\cos \theta}\). So, our expression becomes: $$\sec^2 \theta$$ Thus, we have proven the trigonometric identity: $$\tan^2 \theta + 1 = \sec^2 \theta$$

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Key Concepts

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

Tangent Function
The tangent function is one of the core trigonometric functions used to describe relationships in right-angled triangles. It is commonly represented as \( \tan \theta \), where \( \theta \) is the angle in question. The tangent of an angle is defined as the ratio of the opposite side to the adjacent side in a right-angled triangle. In terms of sine and cosine, the tangent function can be expressed as:
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \]
This relationship allows us to express the tangent through functions that may be easier to handle in various trigonometric manipulations.
  • This conversion is useful, especially when dealing with trigonometric identities.
  • By expressing \( \tan \theta \) in this way, the tangent function highlights the interdependence between different trigonometric functions.
Secant Function
The secant function is another fundamental trigonometric function, denoted as \( \sec \theta \). It is defined as the reciprocal of the cosine function:
\[ \sec \theta = \frac{1}{\cos \theta} \]
This function comes into play particularly in identities and problems where the cosine of an angle is given, and you seek its reciprocal.
  • Such transformation is often used to solve more complex trigonometric identities or equations.
  • In comparison with the cosine function, secant is less commonly taught initially but is crucial for comprehensive trigonometric knowledge.
Given our problem, knowing that \( \sec^2 \theta \) appears in the identity serves as a key point in transitioning from one function form to the other.
Pythagorean Identity
The Pythagorean identity is a vital tool in trigonometry, stating that:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
This identity arises from the Pythagorean Theorem and is fundamental in verifying and manipulating trigonometric equations. It often serves as the bridge to simplify expressions in terms of different trigonometric functions.
  • In the given solution, the Pythagorean identity helps in transforming \( \frac{\sin^2 \theta + \cos^2 \theta}{\cos^2 \theta} \) to \( \sec^2 \theta \).
  • Understanding this identity can unlock solutions to a variety of trigonometric problems.
  • The simplicity of the identity belies its importance and wide applicability across mathematics.
Recognizing and using this identity effectively is essential for proving identities, simplifying expressions, and comprehending deeper theoretical concepts in trigonometry.

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Most popular questions from this chapter

Kelly has finished a picnic on an island that is \(200 \mathrm{m}\) off shore (see figure). She wants to return to a beach house that is \(600 \mathrm{m}\) from the point \(P\) on the shore closest to the island. She plans to row a boat to a point on shore \(x\) meters from \(P\) and then jog along the (straight) shore to the house. a. Let \(d(x)\) be the total length of her trip as a function of \(x .\) Graph this function. b. Suppose that Kelly can row at \(2 \mathrm{m} / \mathrm{s}\) and jog at \(4 \mathrm{m} / \mathrm{s}\). Let \(T(x)\) be the total time for her trip as a function of \(x\). Graph \(y=T(x)\) c. Based on your graph in part (b), estimate the point on the shore at which Kelly should land in order to minimize the total time of her trip. What is that minimum time?

Use analytical methods to find the following points of intersection. Use a graphing utility only to check your work. Find the point(s) of intersection of the parabola \(y=x^{2}+2\) and the line \(y=x+4\)

Modify Exercise 84 and use property \(\mathrm{E} 2\) for exponents to prove that \(\log _{b}(x / y)=\log _{b} x-\log _{b} y\).

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. \(\sin (a+b)=\sin a+\sin b\) b. The equation \(\cos \theta=2\) has multiple real solutions. c. The equation \(\sin \theta=\frac{1}{2}\) has exactly one solution. d. The function \(\sin (\pi x / 12)\) has a period of 12 e. Of the six basic trigonometric functions, only tangent and cotangent have a range of \((-\infty, \infty)\) f. \(\frac{\sin ^{-1} x}{\cos ^{-1} x}=\tan ^{-1} x\) g. \(\cos ^{-1}(\cos (15 \pi / 16))=15 \pi / 16\) h. \(\sin ^{-1} x=1 / \sin x\)

Determine whether the graphs of the following equations and functions have symmetry about the \(x\) -axis, the \(y\) -axis, or the origin. Check your work by graphing. $$f(x)=2|x|$$
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