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Chapter 8: Problem 42

Establish each identity. $$9 \sec ^{2} \theta-5 \tan ^{2} \theta=5+4 \sec ^{2} \theta$$

Short Answer

Expert verified
Identity established: \[9 \sec ^{2} \theta - 5 \tan ^{2} \theta = 5 + 4 \sec ^{2} \theta\].

Step by step solution

01

- Write the given identity

Start with the given identity: \[9 \sec ^{2} \theta - 5 \tan ^{2} \theta = 5 + 4 \sec ^{2} \theta\]
02

- Recall the trigonometric identities

Use the trigonometric identity: \[1 + \tan ^{2} \theta = \sec ^{2} \theta\]. This can be rearranged to: \[\tan ^{2} \theta = \sec ^{2} \theta - 1\]
03

- Substitute the \tan ^{2} \theta

Replace \tan ^{2} \theta in the original equation with \sec ^{2} \theta - 1: \[9 \sec ^{2} \theta - 5(\sec ^{2} \theta - 1)\]. Simplify inside the parentheses: \[9 \sec ^{2} \theta - 5 \sec ^{2} \theta + 5\]
04

- Combine like terms

Combine the \sec ^{2} \theta terms: \[9 \sec ^{2} \theta - 5 \sec ^{2} \theta + 5 = 4 \sec ^{2} \theta + 5\]. Now the equation is: \[4 \sec ^{2} \theta + 5 = 4 \sec ^{2} \theta + 5\].
05

- Verify the identity

Both sides of the equation are equal, thus the original equation is established as 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, denoted as \(\sec \theta\), is a fundamental trigonometric function. It is defined as the reciprocal of the cosine function: \(\sec \theta = \frac{1}{\cos \theta}\). This means that the secant function is related to the ratio of the hypotenuse to the adjacent side in a right triangle.
When working with secant, remember these key points:
  • \(\sec \theta\) is undefined when \(\cos \theta = 0\).
  • The secant function has a period of \(2\pi\), the same as the cosine function.
  • \(\sec \theta\) can be greater than or equal to 1 or less than or equal to -1.
In identities and expressions, \(\sec^2 \theta\) often appears, sometimes leading to the relationship \(1 + \tan^2 \theta = \sec^2 \theta\). This identity is used frequently in simplifying and verifying trigonometric expressions.
Tangent Function
The tangent function, denoted as \(\tan \theta\), is another primary trigonometric function. It is defined as the ratio of the sine function to the cosine function: \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). This means that the tangent function represents the ratio of the opposite side to the adjacent side in a right triangle.
Key points to remember about the tangent function:
  • \(\tan \theta\) is undefined when \(\cos \theta = 0\).
  • The tangent function has a period of \(\pi\).
  • \(\tan \theta\) varies smoothly and can take any real value.
In trigonometric identities, \(\tan^2 \theta\) frequently appears, often requiring simplification adjustments like using the identity \(\tan^2 \theta = \sec^2 \theta - 1\). This identity helps to bridge between different trigonometric functions and simplifies complex expressions effectively.
Simplifying Trigonometric Expressions
Simplifying trigonometric expressions often involves using fundamental identities to rewrite and reduce expressions. In the provided exercise, we used the identity \(\sec^2 \theta = 1 + \tan^2 \theta\) to simplify the expression.
Steps in simplification usually include:
  • Identifying relevant trigonometric identities.
  • Substituting and rearranging terms accordingly.
  • Combining like terms to simplify the overall expression.
For example, in the given problem, we rewrote \(\tan^2 \theta\) in terms of \(\sec^2 \theta\) and simplified the equation:
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