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Chapter 7: Problem 97

Verify that each equation is an identity. $$\frac{2}{1+\cos x}-\tan ^{2} \frac{x}{2}=1$$

Short Answer

Expert verified
The given equation is an identity.

Step by step solution

01

- Use a Trigonometric Identity

Start by using the trigonometric identity for \(\cos 2\theta\): \(\cos x = 1 - 2\sin^2 \frac{x}{2}\). Here, \(2\theta = x\), so substitute into \(\frac{2}{1 + \cos x}\)
02

- Substitute the Identity

Substitute \(\cos x = 1 - 2\sin^2 \frac{x}{2}\) into \(\frac{2}{1 + \cos x}\): \[\frac{2}{1 + (1 - 2\sin^2 \frac{x}{2})} = \frac{2}{2 - 2\sin^2 \frac{x}{2}} = \frac{2}{2(1 - \sin^2 \frac{x}{2})} = \frac{1}{1 - \sin^2 \frac{x}{2}}\]
03

- Simplify Using Algebra

Simplify \(\frac{1}{1 - \sin^2 \frac{x}{2}}\): \[\frac{1}{1 - \sin^2 \frac{x}{2}} = \frac{1}{\cos^2 \frac{x}{2}}= \sec^2 \frac{x}{2}\]
04

- Simplify \tan^2 \frac{x}{2}\

Recall that \sec^2 \theta= 1 + \tan^2 \theta\. Therefore, the equation can be written as: \[ \sec^2 \frac{x}{2} - \tan^2 \frac{x}{2} = 1\]
05

- Verify the Identity

Combine the simplified form into the original expression: \[ \frac{2}{1 + \cos x } = \sec^2 \frac{x}{2}, \sec^2 \frac{x}{2} - \tan^2 \frac{x}{2}= 1\] This confirms the original identity.

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Key Concepts

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

Cosine Double-Angle Identity
The cosine double-angle identity is a key concept in trigonometry. It expresses cosine of a double angle in terms of sine or cosine of a single angle. The formula is:
\[ \cos(2\theta) = 1 - 2 \sin^2(\theta) \]
In the given exercise, this identity is used to simplify expressions involving \cos(x)\. Specifically, by substituting \cos(x) = 1 - 2 \sin^2 \left( \frac{x}{2} \right)\ when x is twice an angle (2胃).
This helps to convert the double-angle cosine term into a format that's easier to manage and simplify.
Secant
The secant function is another crucial concept. It relates to the cosine function and can simplify many trigonometric identities. Secant is defined as:
\[ \sec(\theta) = \frac{1}{\cos(\theta)} \]
In the solution, secant was used through the simplification:
  • Starting with the term \frac{1}{1 - \sin^2 \left( \frac{x}{2} \right)}\,
  • Recognize that \1 - \sin^2 \left( \frac{x}{2} \right) = \cos^2 \left( \frac{x}{2} \right)\,
  • So the term simplifies to \sec^2 \left( \frac{x}{2} \right)\.
This shows the power of recognizing and using secant in trigonometric identities to simplify expressions.
Tangent
Tangent is a fundamental trigonometric function related to sine and cosine. It's defined as:
\[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \]
The secant and tangent are related through the identity:
\[ \sec^2(\theta) = 1 + \tan^2(\theta) \]
This relationship was crucial in verifying the given identity. Here's how it was applied:
  • When the expression simplified to \sec^2 \left( \frac{x}{2} \right) \,
  • Recognize from the identity that \sec^2 \left( \frac{x}{2} \right) - \tan^2 \left( \frac{x}{2} \right) = 1 \.
This confirmed that the original equation is, indeed, an identity.

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Most popular questions from this chapter

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