Problem 762 Prove the identity \(\csc 2 \mat... [FREE SOLUTION]

Chapter 35: Problem 762

Prove the identity \(\csc 2 \mathrm{x}=(\csc \mathrm{x} / 2 \cos \mathrm{x})\).

Short Answer

Expert verified

To prove the trigonometric identity \(\csc 2x = \frac{\csc x}{2\cos x}\), we follow these steps: 1. Write the double-angle identity for \(\sin 2x\): \(\sin 2x = 2\sin x\cos x\). 2. Find the reciprocal of \(\sin 2x\) to obtain \(\csc 2x = \frac{1}{\sin 2x}\). 3. Substitute the double-angle identity for \(\sin 2x\): \(\csc 2x = \frac{1}{2\sin x\cos x}\). 4. Rewrite the expression in terms of \(\csc x\) and \(\cos x\): \(\csc 2x = \frac{\csc x}{2\cos x}\). Thus, the identity is proven.

Step by step solution

Write the Double Angle Identity

We start by writing the double-angle identity for \(\sin 2x\), which is given by \(\sin 2x = 2\sin x\cos x\).

Find \(\csc 2x\) using its Reciprocal Identity

Next, we find the reciprocal of \(\sin 2x\) to obtain \(\csc 2x\). The reciprocal identity for \(\csc\) is given by \(\csc x = \frac{1}{\sin x}\). Thus, we have \[ \csc 2x = \frac{1}{\sin 2x}. \]

Substitute the Double Angle Identity into \(\csc 2x\)

Now, substitute the double-angle identity for \(\sin 2x\) from Step 1 into the expression for \(\csc 2x\): \[ \csc 2x = \frac{1}{2\sin x\cos x}. \]

Rewrite in terms of \(\csc x\) and \(\cos x\)

Finally, rewrite the expression in terms of \(\csc x\) and \(\cos x\) using the reciprocal identity for \(\csc x\) again: \[ \csc 2x = \frac{1}{2\sin x\cos x} = \frac{\csc x}{2\cos x}. \] Thus, we have shown that \(\csc 2x = \frac{\csc x}{2\cos x}\).

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.

Double Angle Identity

The double angle identity is a fundamental concept in trigonometry used to simplify complex trigonometric expressions or evaluate exact values. Specifically, the double angle identity for sine is written as \( \sin 2x = 2\sin x\cos x \). This identity is very useful in solving trigonometric proofs and problems because it relates the sine of a double angle, \( 2x \), to the sine and cosine of the single angle, \( x \).

Understanding this allows us to transform expressions and identify equivalent forms.

The identity provides a link between linear trigonometric functions and their composite angles.
Being able to rewrite expressions using this identity aids in further algebraic simplifications or transformations.

Remember, double angle identities are not only limited to sine, but also exist for cosine and tangent, each offering a different form of equation to work with.

Reciprocal Identity

Reciprocal identities in trigonometry help us express trigonometric functions as the reciprocals of other functions. A classic example is the cosecant function, \( \csc x \), which is the reciprocal of the sine function:

\[ \csc x = \frac{1}{\sin x} \]

Using reciprocal identities is crucial when simplifying expressions or proving other identities.

They allow expressions to be manipulated into forms that might be easier to work with or compare.
Understanding these identities helps provide alternative methods to approach problems.

In our exercise, recognizing the relationship between \( \csc 2x \) and \( \sin 2x \), we effectively used the reciprocal identity to develop our proof.

Cosecant Function

The cosecant function, denoted as \( \csc x \), is one of the basic trigonometric functions closely related to the sine function. Defined as the reciprocal of sine, it has the expression \( \csc x = \frac{1}{\sin x} \).

The cosecant is significantly used in trigonometric proofs and calculations, such as in discovering new identities or simplifying expressions. Some properties of the cosecant to remember include:

\( \csc x \) is undefined whenever \( \sin x = 0 \), leading to vertical asymptotes in its graph.
It shares periodicity with the sine function, repeating every \( 2\pi \).

By understanding and identifying when to use \( \csc x \) instead of \( \sin x \), we can more effectively manipulate trigonometric expressions.

Trigonometric Proofs

Trigonometric proofs involve showing that two expressions, typically containing trigonometric functions, are equivalent. To construct a successful trigonometric proof, fundamental identities, like the double angle and reciprocal identities, are commonly utilized.

When working on a proof, key strategies include:

Simplifying one or both sides of the equation using trigonometric identities.
Substituting known identities to create equivalence between expressions.

In our example exercise, we demonstrated the identity \( \csc 2x = \frac{\csc x}{2\cos x} \) by leveraging the double angle identity \( \sin 2x = 2 \sin x \cos x \) and the reciprocal identity \( \csc x = \frac{1}{\sin x} \). These steps show how systematically applying these identities helps in proving equivalences in trigonometry.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Prove the identity \(\csc 2 \mathrm{x}=(\csc \mathrm{x} / 2 \cos \mathrm{x})\).

Short Answer

Step by step solution

Write the Double Angle Identity

Find \(\csc 2x\) using its Reciprocal Identity

Substitute the Double Angle Identity into \(\csc 2x\)

Rewrite in terms of \(\csc x\) and \(\cos x\)

Key Concepts

Double Angle Identity

Reciprocal Identity

Cosecant Function

Trigonometric Proofs

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Calculus

Probability and Statistics

Theoretical and Mathematical Physics

Geometry

Decision Maths

Study anywhere. Anytime. Across all devices.