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Chapter 8: Problem 80

Establish each identity. $$\frac{\sec ^{2} v-\tan ^{2} v+\tan v}{\sec v}=\sin v+\cos v$$

Short Answer

Expert verified
Express all terms as a fraction, using reciprocal identities verify the solution by simplifying Doulistically as a combination of terms of tangent by it self.

Step by step solution

01

Recall Trigonometric Identities

Recall the identity \(\tan^2 v + 1 = \frac{1}{\tan^2 v} = \frac{\tan^2 v}{\tan^2 v}\). This helps in simplifying the given expression.
02

Expression Simplification

Rewrite \( \frac{\tan^2 v + 1 - \tan^2 v + \tan v}{\tan v} \) as \( \frac{1 + \tan v - \tan^2 v}{\tan v} \)
03

Convert Terms to Sine and Cosine

Using identity for \(\tan v = \frac{\tan v}{\tan v} \), convert the terms into sines and cosines: \( \frac{\frac{\tan^{2}v}{\tan^{2}v}}{\tan^{2}v } \)
04

Combine Fractions

Cancel out the fraction \( \frac{1 \tan v}{\frac{1+\tan v-\tan^2 v}{\tan^2 v}} \)
05

Simplify and Verify

Simplify the expression: \( 1+\tan ^{2}v+\tan ^{2} v-\frac{\tan^2 v+\tan v}{\tan v}+\frac{1}{1 \tan v }+\frac{1}{\tan v}-\frac{\tan^2 v}{\tan^2 v +\tan v+\frac{\tan v}{\tan^2 v}} \)

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Key Concepts

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

trigonometric simplification
Trigonometric simplification is a process used to make complex trigonometric expressions more manageable. To simplify a trigonometric expression, we often employ various fundamental identities like the Pythagorean identities, angle sum and difference identities, and reciprocal identities. In our example, we utilize the Pythagorean identity: \[ \tan^2 v + 1 = \frac{1}{\tan^2 v} \] By recognizing and applying these identities, we can convert and combine terms to make the expression simpler. For instance, if you encounter the term \( \tan^2 v + 1 \), recalling that this is equivalent to \( \frac{1}{\tan^2 v} \) helps break down and simplify the original trigonometric equation.
sine and cosine functions
Sine and cosine functions are the core of all trigonometric concepts. They help convert more complex trigonometric functions, making it easier to simplify and solve equations. For example, the tangent function is expressed using sine and cosine functions: \( \tan v = \frac{\tan v}{\tan v} = \frac{\tan^2 v}{\tan^2 v} \).Using this property, we can rewrite the expression in terms of sine and cosine, facilitating simpler forms for manipulation. For instance, when you encounter \( \tan v \), breaking it down into \( \frac{\tan v}{\tan v} \) allows one to handle the expression with the more fundamental sine and cosine functions.
algebraic manipulation
Algebraic manipulation involves operations like addition, subtraction, multiplication, and division to simplify or solve an equation. For trigonometric problems, manipulating fractions and terms can be particularly helpful. In our exercise, an expression like \( \frac{1 + \tan v - \tan^2 v}{\tan v} \) requires cancelling out terms and simplifying fractions. By reorganizing and combining terms properly, a complicated equation becomes solvable. The key steps encompass:
  • Combining like terms
  • Canceling out common factors
  • Converting expressions using known identities
Regular practice and familiarity with these tactics will enhance your ability to handle complex trigonometric equations efficiently.

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Most popular questions from this chapter

If \(x=2 \tan \theta,\) express \(\sin (2 \theta)\) as a function of \(x\)

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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Solve each equation on the interval \(0 \leq \theta<2 \pi\). $$ \cot \theta+\csc \theta=-\sqrt{3} $$

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