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Chapter 35: Problem 755

Show that \(\sec ^{2} \theta-\tan ^{2} \theta=1\) is an identity.

Short Answer

Expert verified
Given the expression \(\sec^2 \theta - \tan^2 \theta\), we substitute the definitions of secant and tangent functions: \(\sec \theta = \frac{1}{\cos \theta}\) and \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). The expression becomes \(\frac{1}{\cos^2 \theta} - \frac{\sin^2 \theta}{\cos^2 \theta}\). Combining the fractions and using the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\), we obtain \(\frac{\cos^2 \theta}{\cos^2 \theta}\), which simplifies to 1. Therefore, \(\sec^2 \theta - \tan^2 \theta = 1\) is an identity.

Step by step solution

01

Recall Definitions of Secant and Tangent Functions

The secant function is defined as the reciprocal of the cosine function, so \(\sec \theta = \frac{1}{\cos \theta}\). The tangent function is defined in terms of sine and cosine functions, so \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).
02

Substitute Definitions into Expression

Replace secant and tangent functions in the given expression \(\sec^2 \theta - \tan^2 \theta\) with their respective definitions: \[ \left(\frac{1}{\cos \theta}\right)^2 - \left(\frac{\sin \theta}{\cos \theta}\right)^2 \]
03

Simplify Expression

Simplify the expression by squaring the terms and finding a common denominator: \[ \frac{1}{\cos^2 \theta} - \frac{\sin^2 \theta}{\cos^2 \theta} \] Now, combine the fractions using the common denominator \(\cos^2 \theta\): \[ \frac{1 - \sin^2 \theta}{\cos^2 \theta} \]
04

Recall the Pythagorean Identity

Recall the fundamental Pythagorean identity relating sine and cosine functions: \(\sin^2 \theta + \cos^2 \theta = 1\). Using this identity, we can rewrite the expression in Step 3 as: \[ \frac{\cos^2 \theta}{\cos^2 \theta} \]
05

Simplify Result and Conclude

Lastly, simplify our expression by dividing the numerator and denominator by the same term: \[ \frac{\cos^2 \theta}{\cos^2 \theta} = 1 \] Thus, we have proven that \(\sec^2 \theta - \tan^2 \theta = 1\) is an identity.

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Key Concepts

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

Secant Function
The secant function, often abbreviated as \( \sec \theta \), is one of the six fundamental trigonometric functions and is closely tied to the concept of the cosine function. The secant function is defined as the reciprocal of the cosine function. In mathematical terms, this is expressed as \( \sec \theta = \frac{1}{\cos \theta} \).
This means that wherever the cosine value equals zero, the secant value is undefined, because division by zero is not allowed.
  • Secant is related to the adjacent side and hypotenuse in a right-angled triangle.
  • In the unit circle, the secant functions extend beyond the unit circle's radius to represent larger angles.
  • Understanding the secant function helps in analyzing complex wave forms and oscillations.
Being able to express secant in terms of cosine is crucial for simplifying trigonometric expressions, as we see in this exercise.
Tangent Function
The tangent function, symbolized as \( \tan \theta \), plays a vital role in trigonometry, representing the ratio between the opposite side and adjacent side in a right triangle. It's also linked to sine and cosine through the formula \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).
This function uniquely bridges both sine and cosine, allowing us to analyze angles and angular paths comprehensively.
  • The tangent function becomes undefined whenever the cosine function is zero.
  • Tangent offers a smooth transition in function values from zero degrees up to negative infinity at 90 degrees.
  • It is periodic, repeating its values in cycles of \( \pi \) radians or 180 degrees.
Understanding the tangent function is useful in trigonometric identities and solving equations, as demonstrated in transforming \( \tan^2 \theta \) in this exercise.
Pythagorean Identity
The Pythagorean identity is a fundamental relation in trigonometry. It describes the squares of the sine and cosine functions, leading to the identity: \( \sin^2 \theta + \cos^2 \theta = 1 \).
This relationship is crucial as it provides a link between the trigonometric functions, allowing us to transform and solve various equations.
  • It originates from the Pythagorean theorem in geometry, applied to unit circles.
  • This identity is used to derive other important trigonometric identities, such as \( \sec^2 \theta = 1 + \tan^2 \theta \).
  • Using it can simplify problems involving trigonometric sums and differences.
As seen in the exercise, applying the Pythagorean identity helps reduce the given expression to a simpler form, verifying it as an identity.
Cosine Function
The cosine function, denoted as \( \cos \theta \), is one of the primary trigonometric functions that describe the relation between angles and ratios in a triangle. Specifically, it represents the ratio of the length of the adjacent side to the hypotenuse in a right-angled triangle.
Cosine functions are pivotal when analyzing wave behaviors and circular motion.
  • Cosine values range between -1 and 1 and vary cyclically with the angle.
  • It defines the horizontal displacement of points on the unit circle.
  • Cosine is an even function, meaning \( \cos(-\theta) = \cos(\theta) \).
Recognizing its patterns and properties aids in understanding how identities like \( \sec^2 \theta - \tan^2 \theta = 1 \) hold true, as cosine forms the foundation for both secant and tangent calculations.

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Most popular questions from this chapter

Prove the identity \(\tan (\pi / 4+\theta / 2)=\sec \theta+\tan \theta\).

If \(\mathrm{u}\) and \(\mathrm{v}\) are two numbers such that \(\mathrm{u}+\mathrm{v}=(1 / 2) \pi\), show that \(\sin ^{2} \mathrm{u}+\sin ^{2} \mathrm{v}=1\)

Prove that \(\cos ^{4} \mathrm{~B}-\sin ^{4} \mathrm{~B}=\cos ^{2} \mathrm{~B}-\sin ^{2} \mathrm{~B}\) is an identity.

Determine the non-negative values of \(x\) less than \(2 \pi\) for which \(2 \cos ^{2} x+\sin x-2>0\)

Show that \(\tan ^{2} t+1=\sec ^{2} t\).
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