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Chapter 20: Problem 23

Simplify the given expressions. $$2 \cos ^{2} \frac{1}{2} x-1$$

Short Answer

Expert verified
The expression simplifies to \(\cos x\).

Step by step solution

01

Identify the Trigonometric Identity

Recognize that the expression \(2 \cos^2 \frac{1}{2} x - 1\) resembles a trigonometric identity. The formula related to this expression is \(\cos 2x = 2\cos^2 x - 1\).
02

Substitute the Identity

Substitute the angle \(x\) in the identity \(\cos 2x = 2\cos^2 x - 1\) with \(\frac{1}{2} x\). This gives us \(\cos x = 2\cos^2 \frac{1}{2} x - 1\).
03

Rewrite the Original Expression

Using the identity substitution from Step 2, rewrite \(2 \cos^2 \frac{1}{2} x - 1\) as \(\cos x\).
04

Final Simplified Expression

The expression \(2 \cos^2 \frac{1}{2} x - 1\) simplifies directly to \(\cos x\).

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Key Concepts

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

Simplifying Expressions
Simplifying expressions in trigonometry often involves manipulating formulas to achieve a more straightforward form. Our target is to transform a complex or unfamiliar expression into something recognizable.
  • Always look for known trigonometric identities that may apply.
  • The idea is to match parts of the expression to these identities.

In our example, we need to simplify the expression \(2 \cos^2 \frac{1}{2} x - 1\). By recognizing a connection to a standard identity, we can transform it into a simpler form. Look for expressions that match well-known formulas, such as Pythagorean identities or angle sum and difference identities. This eventually simplifies our work and leads us to a more familiar form, often making calculations more straightforward.
Cosine Formula
The cosine formula is a powerful tool in trigonometry. It frequently relates to the cosine of double angles, which is particularly useful. In this context, the formula \( \cos 2x = 2 \cos^2 x - 1\) helps us to solve exercises like the one we are discussing.
  • This specific formula shows how the cosine of a double angle is connected to the square of the cosine of the base angle.
  • It presents a unique way to simplify expressions involving squared cosine terms.

In our problem, we used this identity by replacing \( x \) with \( \frac{1}{2}x\), thus converting the expression \(2 \cos^2 \frac{1}{2} x - 1\) into \( \cos x \). Learning to recognize the cosine formula automatically can make it easier to simplify and solve trigonometric expressions.
Half-Angle Identity
Half-angle identities are very useful in trigonometry, especially when dealing with transformations and simplifications. They express trigonometric functions based on half of a given angle, helping us to connect otherwise complex expressions to more basic angles.
  • These identities are particularly important when working with expressions or equations involving angles like \(\frac{1}{2}x\) or similar fractions of angles.
  • They extend the usefulness of trigonometric identities, such as the double angle identities, allowing for broader applications.
In structural terms, the half-angle identity used here rearranged the equation \( \cos x = 2 \cos^2 \frac{1}{2} x - 1 \). The transformation allowed us to simplify this complex form into a straightforward expression \( \cos x \). Understanding and applying half-angle identities can robustly enhance problem-solving strategies in trigonometry.

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Most popular questions from this chapter

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