Problem 23 - Algebra 2

Question

Verify that each of the following is an identity. $$ 2 \cos ^{2} \frac{x}{2}=1+\cos x $$

Step-by-Step Solution

Answer

The expression is an identity, matching a known trigonometric formula.

Step 1: Recall Trigonometric Identity

Recall the double angle identity for cosine: \[ \cos 2\theta = 2 \cos^2 \theta - 1 \] From this, we can express \( 2\cos^2 \theta \) as:\[ 2 \cos^2 \theta = 1 + \cos 2\theta \]

Step 2: Apply Substitution

Set \( \theta = \frac{x}{2} \). This makes \( 2\theta = x \).Thus, the identity becomes:\[ 2\cos^2 \frac{x}{2} = 1 + \cos x \]

Step 3: Verify the Given Identity

Since after substitution \( 2 \cos^2 \frac{x}{2} = 1 + \cos x \) directly matches our known identity \( 2 \cos^2 \theta = 1 + \cos 2\theta \) with \( \theta = \frac{x}{2} \), the given expression is an identity.

Key Concepts

Double Angle Formulas

Understanding double angle formulas is a crucial part of trigonometry. These formulas, especially the cosine double angle formula, help simplify complex trigonometric expressions and verify identities.

The double angle formula for cosine is given by: \[\cos 2\theta = 2 \cos^2 \theta - 1\] This formula allows you to express the cosine of double angles in terms of the cosine of single angles.

When you know the cosine of an angle, you can easily find the cosine of twice that angle using this identity. This is particularly useful in calculus, physics, and many fields where trigonometric equations appear.

• To deduce other forms or verify identities, the formula can be rearranged as:
  \(2 \cos^2 \theta = 1 + \cos 2\theta\)
• Notice how this allows substitution to prove various trigonometric identities.

Understanding these connections helps build a deeper understanding of trigonometric problems.

Cosine Function

The cosine function is one of the foundational trigonometric functions. It establishes relationships in triangles and models periodic phenomena.

It is represented as \( \cos(x) \) and describes the x-coordinate of a point on the unit circle. For any angle \(x\), \( \cos(x)\) tells us how far left or right we moved from the circle's center.

Being cyclical, the cosine function has a period of \(2\pi\). It repeats its values over this interval, displaying its wave-like characteristics.

• Its maximum value is 1, and its minimum is -1, reached at \(\cos(0) = 1\) and \(\cos(\pi) = -1\) respectively.
• The function starts at 1, descends to -1 by \(\pi\), then returns to 1 by \(2\pi\).

Utilizing the cosine function can simplify the understanding and solution of many geometric and trigonometric problems.

Trigonometric Verification

Trigonometric verification involves proving that two trigonometric expressions are equivalent, verifying the identity of one expression by transforming it into another. This is a fundamental task in trigonometry, allowing students to deepen their understanding of trigonometric relationships.

To verify an identity:

• Start by recognizing known identities, such as the double angle or Pythagorean identities.
• Substitute or transform parts of the expression using these identities.
• Ensure both sides of the equation match after simplification.

A common method is substitution, as we see with the identity \(2 \cos^2 \frac{x}{2} = 1 + \cos x\). By setting \( \theta = \frac{x}{2}\), we use the double angle formula to show equivalency. This approach confirms the transformation of more complex expressions into simpler forms, validating the identity through logical steps.