Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 8: Problem 79
Establish each identity. $$\frac{(\sec v-\tan v)^{2}+1}{\csc v(\sec v-\tan v)}=2 \tan v$$
Short Answer
Expert verified
The given expression simplifies to \[2 \tan v\].
Step by step solution
01
Simplify the Numerator
Start with the expression \[(\sec v - \tan v)^2 + 1\]. Notice that \(\sec^2 v - \tan^2 v = 1\). So, \[(\sec v - \tan v)^2 \] can be expanded as \[(\sec^2 v - 2\sec v \tan v + \tan^2 v)\]. Adding 1 to this expression results in \[(\sec^2 v - 2\sec v \tan v + \tan^2 v + 1)\].
02
Use Trigonometric Identities
Recall the Pythagorean identities: \[\sec^2 v = 1 + \tan^2 v\] and \[\csc^2 v = 1 + \cot^2 v\]. Substituting \[\sec^2 v = 1 + \tan^2 v\] into the previous result gives us: \[(1 + \tan^2 v - 2\sec v \tan v + \tan^2 v + 1) = 2 + 2\tan^2 v - 2\sec v \tan v\].
03
Simplify the Result
Notice that \[2 + 2 \tan^2 v - 2 \sec v \tan v \] can be factored as \[2(1 + \tan^2 v - \sec v \tan v)\].
04
Simplify the Denominator
The denominator \[\csc v (\sec v - \tan v)\] can be rewritten using \[\csc v = \frac{1}{\sin v}\], which gives \[(\frac{1}{\sin v})(\sec v - \tan v) = \frac{\sec v - \tan v}{\sin v}\].
05
Combine and Simplify
Combine the simplified numerator and denominator: \[(2(1 + \tan^2 v - \sec v \tan v)) / (\frac{\sec v - \tan v}{\sin v})\]. Notice that \[\sec v = \frac{1}{\cos v}\] and \[\tan v = \frac{\sin v}{\cos v}\]. The expression simplifies to: \[(2(1 + \tan^2 v - \sec v \tan v)) * (\frac{\sin v}{\sec v - \tan v})\].
06
Simplify Further
Use the trigonometric identities again to simplify further. This leads to \[(2) * \tan v = 2 \tan v\].
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Pythagorean identities
In trigonometry, a key set of identities is known as the Pythagorean identities. These identities relate the squares of the main trigonometric functions. The most fundamental identity is \( \sin^2 \theta + \cos^2 \theta = 1 \). From this, we can derive other identities. For instance, by dividing each term by \( \cos^2 \theta \), we get \( 1 + \tan^2 \theta = \sec^2 \theta \). Similarly, dividing by \( \sin^2 \theta \) results in \( 1 + \cot^2 \theta = \csc^2 \theta \). These identities are particularly useful in simplifying complex trigonometric expressions.
secant function
The secant function, denoted as \( \sec \), is the reciprocal of the cosine function. This means: \[ \sec \theta = \frac{1}{\cos \theta} \]. Because it鈥檚 the reciprocal of cosine, when \( \cos \theta \) approaches zero, \( \sec \theta \) increases rapidly. The secant function also plays an important role in the Pythagorean identity mentioned earlier, \( \sec^2 \theta = 1 + \tan^2 \theta \), helping to relate it to the tangent function and simplifying expressions containing trigonometric functions.
tangent function
The tangent function, denoted as \( \tan \), is the ratio of the sine function to the cosine function: \[ \tan \theta = \frac{\sin \theta}{\cos \theta} \]. When \( \theta \) is close to \( 90^\circ \) or \( 270^\circ \), \( \cos \theta \) approaches zero, causing \( \tan \theta \) to increase or decrease without bound. Tangent has some interesting properties. For example, it repeats every \( 180^\circ \). As presented in the exercise, the identity \( \sec^2 \theta = 1 + \tan^2 \theta \) shows the relationship between secant and tangent, which helps to simplify various trigonometric expressions.
simplifying trigonometric expressions
Simplifying trigonometric expressions often involves using various trigonometric identities and properties, like the Pythagorean identities, reciprocal identities, and quotient identities. In the given exercise, we applied the identity \( \sec^2 v = 1 + \tan^2 v \) to transform the given expression. By recognizing patterns and substituting equivalent forms, we can break down complex expressions into simpler ones. Here's a quick tip: always look for opportunities to use identities to combine or cancel terms, which can significantly reduce the complexity of the problem.
