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Chapter 6: Problem 54

Verify each of the trigonometric identities. $$\frac{1}{1-\cos x}+\frac{1}{1+\cos x}=2 \csc ^{2} x$$

Short Answer

Expert verified
The identity \( \frac{1}{1-\cos x}+\frac{1}{1+\cos x}=2 \csc^2 x \) is verified.

Step by step solution

01

Simplifying the Left Side

Let's start by finding a common denominator for the left side of the equation: \( \frac{1}{1-\cos x} + \frac{1}{1+\cos x} \). The common denominator is \( (1-\cos x)(1+\cos x) \). So, the left side becomes \( \frac{1(1+\cos x) + 1(1-\cos x)}{(1-\cos x)(1+\cos x)} \).
02

Apply the Difference of Squares Formula

The denominator, \( (1-\cos x)(1+\cos x) \), can be simplified using the difference of squares formula \( a^2 - b^2 = (a-b)(a+b) \). Here, it becomes \( 1^2 - \cos^2 x = 1 - \cos^2 x \). Recall that \( 1 - \cos^2 x = \sin^2 x \).
03

Simplifying the Numerator

The numerator is \( 1 + \cos x + 1 - \cos x = 2 \). So the full expression becomes \( \frac{2}{\sin^2 x} \).
04

Recognizing the Trigonometric Identity

The expression \( \frac{1}{\sin^2 x} \) is equal to \( \csc^2 x \). Therefore, \( \frac{2}{\sin^2 x} \) simplifies to \( 2 \csc^2 x \).
05

Verifying the Identity

Both sides of the original equation, \( \frac{1}{1-\cos x}+\frac{1}{1+\cos x} = 2 \csc^2 x \), simplify to \( 2 \csc^2 x \). Thus, the identity is verified.

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Key Concepts

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

Difference of Squares
In mathematics, the difference of squares is a handy algebraic formula used to simplify expressions. It's given by the formula \( a^2 - b^2 = (a-b)(a+b) \). This equation points out that the difference between two squares is equal to the product of their sum and difference.
  • This concept appears frequently in both algebra and trigonometry.
  • It makes tasks like factoring and simplifying vastly easier.

In our trigonometric identity example, we used the difference of squares formula to simplify \((1 - \cos x)(1 + \cos x)\).
  • Here, \(a = 1\) and \(b = \cos x\), giving us \(1^2 - \cos^2 x\).
  • This simplification reduces to \(1 - \cos^2 x\).
  • In trigonometry, we recognize that \(1 - \cos^2 x = \sin^2 x\).

Using this formula helps us quickly simplify the equation and leads us one step closer to verifying the trigonometric identity.
Cosecant Function
The cosecant function, denoted as \(\csc x\), is a fundamental trigonometric function. It's the reciprocal of the sine function, so \(\csc x = \frac{1}{\sin x}\).
Let's explore some key aspects of the cosecant function:
  • The cosecant function is undefined wherever sine is zero, like at \(x = 0, \pi, 2\pi\), etc.
  • It generally appears in problems involving reciprocal identities in trigonometry.
  • It's useful for verifying identities and solving trigonometric equations.

In the equation from our exercise, the identity \( \frac{1}{\sin^2 x} = \csc^2 x \) was prominent.
  • This is because \(\csc^2 x\) is simply the square of \(\frac{1}{\sin x}\).
  • Recognizing this helped us make sense of the final expression \(2 \csc^2 x\).

The understanding of the cosecant function is crucial to trigonometry and aids in verifying various mathematical identities.
Sine Squared
Sine squared, written as \(\sin^2 x\), is a simple yet crucial trigonometric function. This notation means \( (\sin x)^2 \), often appearing in trigonometry for expressions involving powers of sine.
Key points about \(\sin^2 x\):
  • It's involved in fundamental identities like \(\sin^2 x + \cos^2 x = 1\), known as the Pythagorean identity.
  • When manipulating trigonometric expressions, \(\sin^2 x\) helps simplify and reduce terms.
  • It frequently appears in calculus, especially when working with integrals and derivatives involving trigonometric functions.

In our exercise, we used \(1 - \cos^2 x\) which equals \(\sin^2 x\).
  • This substitution helped us transform the complex trigonometric expression.
  • It allowed us to write the identity in the form \( \frac{2}{\sin^2 x} \), eventually simplifying further.

Understanding \(\sin^2 x\) and its roles in trigonometry is imperative for mastering the verification of identities like the one in our lesson.

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Most popular questions from this chapter

Find the smallest positive values of \(x\) that make the statement true. Give the answer in degrees and round to two decimal places. $$\sec (3 x)+\csc (2 x)=5$$

Find the smallest positive values of \(x\) that make the statement true. Give the answer in degrees and round to two decimal places. $$2 \sin x=\tan x,-\frac{\pi}{3} \leq x \leq \frac{\pi}{3}$$

Refer to the following: The figure below shows the graph of \(y=2 \cos x-\cos (2 x)\) between \(-2 \pi\) and \(2 \pi .\) The maximum and minimum values of the curve occur at the turning points and are found in the solutions of the equation \(-2 \sin x+2 \sin (2 x)=0\). CAN'T COPY THE GRAPH Solve for the coordinates of the turning points of the curve between 0 and \(2 \pi\)

Solve \(16 \sin ^{4} \theta-8 \sin ^{2} \theta=-1\) over \(0 \leq \theta \leq 2 \pi\).

Solve the trigonometric equations exactly on the indicated interval, \(0 \leq x<2 \pi\). $$2 \cot x=\csc x$$
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