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Chapter 20: Problem 34

Prove the given identities. $$2+\frac{\cos 2 \theta}{\sin ^{2} \theta}=\csc ^{2} \theta$$

Short Answer

Expert verified
The identity is proven by demonstrating that both sides simplify to \( \csc^2 \theta \).

Step by step solution

01

Understand the Given Identity

We are given the identity that needs to be proved: \( 2 + \frac{\cos 2\theta}{\sin^2\theta} = \csc^2 \theta \). Our objective is to start from one side of the equation and manipulate it until it resembles the other side.
02

Recall Trigonometric Identities

Remember some key trigonometric identities that might be useful:1. \( \csc \theta = \frac{1}{\sin \theta} \), so \( \csc^2 \theta = \frac{1}{\sin^2 \theta} \).2. The double angle identity for cosine: \( \cos 2\theta = 1 - 2\sin^2 \theta \).
03

Substitute and Simplify

Substitute \( \cos 2\theta = 1 - 2\sin^2 \theta \) into the left-hand side of the equation:\[ 2 + \frac{1 - 2\sin^2\theta}{\sin^2\theta} \].Separate the fraction:\[ 2 + \frac{1}{\sin^2\theta} - 2 \].
04

Combine and Simplify Further

Simplify the expression:\[ 2 + \frac{1}{\sin^2\theta} - 2 = \frac{1}{\sin^2\theta} \]. Thus, \( \frac{1}{\sin^2\theta} = \csc^2 \theta \), which matches the right-hand side of the identity.

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Key Concepts

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

Cosecant Function
Trigonometry involves various functions that relate the angles and ratios of triangles. The cosecant function is one of these functions, typically abbreviated as \( \csc \theta \). The cosecant of an angle \( \theta \) is the reciprocal of the sine function:
  • \( \csc \theta = \frac{1}{\sin \theta} \).
  • Thus, \( \csc^2 \theta = \left( \frac{1}{\sin \theta} \right)^2 = \frac{1}{\sin^2 \theta} \).

These relationships are pivotal when working with trigonometric identities and prove useful when transforming complex expressions. By understanding how the cosecant relates to sine, you can more easily solve various trigonometric problems, especially when identities link different functions together.
Double Angle Identities
Double angle identities are a subset of trigonometric identities that express the trigonometric functions of doubled angles in terms of the functions of the original angle. These identities are particularly useful for simplifying expressions and solving equations when the angle is doubled.

One of the key double angle identities used in this exercise is the formula for cosine:
  • \( \cos 2\theta = 1 - 2\sin^2 \theta \).

By substituting this identity into problems, you can replace \( \cos 2\theta \) with an expression that involves \( \sin \) only, facilitating further algebraic manipulation.
This substitution played a critical role in our exercise, helping simplify the original expression and leading us towards proving the given identity.
Simplifying Expressions
Simplifying expressions is a fundamental skill in mathematics, especially in the field of trigonometry. The aim is to manipulate and reduce expressions to their simplest form, making them easier to solve or understand.

In trigonometry, steps often include using known identities to transform terms. For the given identity, we simplified by:
  • Substituting \( \cos 2\theta \) with \( 1 - 2\sin^2 \theta \).
  • Breaking down complex fractions into simpler parts.
  • Combining like terms to reach an expression equating \( \csc^2 \theta \).

Through these methods, not only did we transform the left side of the given identity into the right side, but we also learned a valuable process for handling similar trigonometric problems. Simplification is key to revealing underlying relationships between functions, often denoted in the form of identities.

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Most popular questions from this chapter

Solve the given problems. The instantaneous electric power \(p\) in an inductor is given by the equation \(p=v i \sin \omega t \sin (\omega t-\pi / 2) .\) Show that this equation can be written as \(p=-\frac{1}{2} v i \sin 2 \omega t\)

Solve the given problems. For voltages \(V_{1}=20 \sin 120 \pi t\) and \(V_{2}=20 \cos 120 \pi t,\) show that \(V=V_{1}+V_{2}=20 \sqrt{2} \sin (120 \pi t+\pi / 4) .\) Use a calcula- tor to verify this result.

In the study of the stress at a point in a bar, the equation \(s=a \cos ^{2} \theta+b \sin ^{2} \theta-2 t \sin \theta \cos \theta\) arises. Show that this equation can be written as \(s=\frac{1}{2}(a+b)+\frac{1}{2}(a-b) \cos 2 \theta-t \sin 2 \theta\)

Solve the given problems. Express \(\sin 3 x\) in terms of \(\sin x\) only.

Solve the given trigonometric equations analytically and by use of a calculator. Compare results. Use values of \(x\) for \(0 \leq x<2 \pi\). $$|\sin x|=\frac{1}{2}$$
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