Question

Simplify the following expressions by writing each one using a single trigonometric function. $$ 4-4 \sin ^{2} \theta $$

Short answer

The expression simplifies to \(2 + 2\cos 2\theta\).

Step-by-step solution

Identify the Formula

Recognize that the expression \( 4 - 4 \sin^2 \theta \) can be simplified using a known trigonometric identity. The double angle identity for cosine, \( \cos 2\theta = 1 - 2 \sin^2 \theta \), will help simplify the expression.

Rewrite Using Known Identity

Notice that \( 1 - 2\sin^2 \theta \) is a part of the identity, so rewrite \( 4 - 4 \sin^2 \theta \) as \( 4(1 - \sin^2 \theta) \). Recognize that \( 1 - \sin^2 \theta = \cos^2 \theta \) using the Pythagorean identity.

Simplify the Expression

Substitute the identity into the expression: \[ 4(1 - \sin^2 \theta) = 4\cos^2 \theta. \] Substitute back again using the double angle identity: \( 1 - \sin^2 \theta \) is \( \cos^2 \theta \), hence the formula becomes \( 4\cos^2 \theta \).

Use Cosine Double Angle Formula

Using the double angle identity \( \cos 2\theta = 1 - 2\sin^2 \theta \), notice that \( 2\cos^2 \theta = 1 + \cos 2\theta \). This simplifies the expression further: \( 4\cos^2 \theta = 2(1 + \cos 2\theta) \) which simplifies to \( 2 + 2\cos 2\theta \).

Key concepts

Cosine Double Angle Identity

The cosine double angle identity is a vital tool for simplifying trigonometric expressions. It states that \( \cos 2\theta = 1 - 2 \sin^2 \theta \) and can also be expressed as \( \cos 2\theta = 2 \cos^2 \theta - 1 \). These formulas show relationships between angles and their trigonometric functions, allowing us to simplify complex expressions.
When you encounter expressions involving squares of sine or cosine, consider using the double angle identities to transform them into simpler forms. This makes the calculations more straightforward, especially when simplifying an expression like our original example.
The process of utilizing this identity often involves recognizing the proper substitution and applying the identity correctly to reduce terms to a single trigonometric function.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions is all about transforming them into their simplest or most accessible form. This process often involves using trigonometric identities to rewrite expressions with fewer terms or with a form that exposes the underlying structure.
In this exercise, we used identities to simplify \( 4 - 4 \sin^2 \theta \) into a more straightforward expression using a cosine function. By recognizing patterns such as \( 1 - 2 \sin^2 \theta \) or the equivalent \( \cos^2 \theta \), the expression was rewritten in a way that reveals its simplicity and underlying cosine double angle identity.
This simplification process reduces errors in calculations and provides a clearer pathway to understanding the behavior of the function.

Pythagorean Identity

The Pythagorean identity is a cornerstone of trigonometry that expresses a fundamental relationship between sine and cosine. It states that \( \sin^2 \theta + \cos^2 \theta = 1 \). This identity is often used to transform trigonometric expressions into different forms.
In our original problem, the Pythagorean identity helped to simplify \( 4 - 4 \sin^2 \theta \). Using the identity \( 1 - \sin^2 \theta = \cos^2 \theta \), we could recognize that the expression simplifies to \( 4 \cos^2 \theta \).
The Pythagorean identity aids in reducing terms involving squares of trigonometric functions, proving instrumental in both the simplification of expressions and in establishing connections between various trigonometric elements.