Problem 4 - A Primer of Real Analysis

Question

Show that for any \(x \in \mathbb{R}\) $$ \sin ^{2}(x)=\frac{1-\cos (2 x)}{2} $$ and $$ \cos ^{2}(x)=\frac{1+\cos (2 x)}{2} $$

Step-by-Step Solution

Answer

Both identities hold based on the cosine double angle formula.

Step 1: Recall Trigonometric Identities

The problem involves showing that two expressions for squared trigonometric functions are equal. To solve this, we use the double angle formulas for sine and cosine. The double angle formulas state: \(\cos(2x) = \cos^2(x) - \sin^2(x)\) and \(\cos(2x) = 2\cos^2(x) - 1\) and \(\cos(2x) = 1 - 2\sin^2(x)\).

Step 2: Express \(\sin^2(x)\) using \(\cos(2x)\)

From the identity \(\cos(2x) = 1 - 2\sin^2(x)\), rearrange to express \(\sin^2(x)\):\[1 - 2\sin^2(x) = \cos(2x)\]Solve for \(\sin^2(x)\):\[2\sin^2(x) = 1 - \cos(2x)\]\[\sin^2(x) = \frac{1 - \cos(2x)}{2}\]This shows the first part of the problem statement.

Step 3: Express \(\cos^2(x)\) using \(\cos(2x)\)

Now, apply the identity \(\cos(2x) = 2\cos^2(x) - 1\). Rearrange to express \(\cos^2(x)\):\[2\cos^2(x) - 1 = \cos(2x)\]Solve for \(\cos^2(x)\):\[2\cos^2(x) = 1 + \cos(2x)\]\[\cos^2(x) = \frac{1 + \cos(2x)}{2}\]This completes the second part of the problem statement.

Key Concepts

Double Angle Formulas

The double angle formulas are important relationships in trigonometry that simplify expressions involving angles. They relate the trigonometric functions of double angles, like \(2x\), to functions of \(x\). To understand these formulas, first remember:

• \(\cos(2x) = \cos^2(x) - \sin^2(x)\)
• \(\cos(2x) = 1 - 2\sin^2(x)\)
• \(\cos(2x) = 2\cos^2(x) - 1\)

It's fascinating that these formulas are interconvertible. By knowing any one form, you can derive the others through simple algebraic manipulations.
For instance, the formula \(\cos(2x) = 1 - 2\sin^2(x)\) allows us to express \(\sin^2(x)\) in terms of \(\cos(2x)\), which is particularly useful when tackling complex trigonometric problems. Understanding how these formulas work can significantly streamline calculations.

Trigonometric Functions

Trigonometric functions are essential in mathematics, especially when working with angles. The most common functions are sine, cosine, and tangent. They have specific relationships and identities that allow them to solve various problems.
The sine function, \(\sin(x)\), represents the y-coordinate of a point on the unit circle, while cosine, \(\cos(x)\), represents the x-coordinate. Both functions are periodic, meaning they repeat their values in a regular pattern.
One important property of these functions is that they have specific identities, such as the Pythagorean identity:

• \( \sin^2(x) + \cos^2(x) = 1 \)

This particular identity lays the groundwork for deriving other identities, like those involving double angles, helping simplify complex expressions. Recognizing and using these functions and identities can deepen your understanding of the interactions between angles and ratios.

Cosine and Sine Relationships

The relationships between cosine and sine are fundamental in solving trigonometric equations. They are interconnected in various ways through identities, one being the double angle formulas.
For example, the identity:

• \( \cos(2x) = 1 - 2\sin^2(x) \)

allows us to express \(\sin^2(x)\) as \(\frac{1 - \cos(2x)}{2}\). This is derived by rearranging the formula to isolate \(\sin^2(x)\).
Similarly, we can find \(\cos^2(x)\) using the formula:

• \( \cos(2x) = 2\cos^2(x) - 1 \)

Rearranging yields \(\cos^2(x) = \frac{1 + \cos(2x)}{2}\).
These transformations not only simplify problems but also highlight the beautiful symmetry and balance within trigonometric identities. Understanding the relationships between sine and cosine aids in recognizing these patterns and applying them effectively in mathematical computations.