Problem 102 - Algebra and Trigonometry

Question

Making Up Your Own Identity If you start with a trigonometric expression and rewrite it or simplify it, then setting the original expression equal to the rewritten expression yields a trigonometric identity. For instance, from Example 1 we get the identity $$ \cos t+\tan t \sin t=\sec t $$ Use this technique to make up your own identity, then give it to a classmate to verify.

Step-by-Step Solution

Answer

The identity is \( \sin^2(x) + \cos^2(x) + \tan^2(x) = 1 + \tan^2(x) \), which can be verified by simplifying both sides.

Step 1: Choose a Trigonometric Expression

Let's start with a trigonometric expression to rewrite or simplify. We'll use \( \sin^2(x) + \cos^2(x) = 1 \), which is a Pythagorean identity. We will manipulate this identity to create a new identity.

Step 2: Introduce New Elements

Rewrite the chosen expression by adding elements like \( \tan(x) \) or \( \sec(x) \). For instance, add \( \tan^2(x) \) to both sides: \[ \sin^2(x) + \cos^2(x) + \tan^2(x) = 1 + \tan^2(x) \].

Step 3: Simplify the Expression

Using the identity \( \tan^2(x) = \sec^2(x) - 1 \), substitute for \( \tan^2(x) \) in the equation: \[ \sin^2(x) + \cos^2(x) + \sec^2(x) - 1 = 1 + \sec^2(x) - 1 \].

Step 4: Simplify Both Sides

Both sides of the equation simplify to this: \( \sin^2(x) + \cos^2(x) + \sec^2(x) - 1 = \sec^2(x) \). Simplifying further: \( 1 + \sec^2(x) - 1 = \sec^2(x) \).

Step 5: Verify the Identity

Since the original expression simplifies to \( \sec^2(x) = \sec^2(x) \), we have successfully shown that our manipulated expression created a valid identity. Each transformation step holds true based on known trigonometric identities.

Key Concepts

Pythagorean Identity

The Pythagorean Identity is one of the most fundamental identities in trigonometry. It states that for any angle \(x\), the sum of the squares of the sine and cosine of that angle equals 1. In mathematical terms, this can be written as \(\sin^2(x) + \cos^2(x) = 1\).
This identity is derived from the Pythagorean Theorem, applied to the unit circle. In a unit circle, the radius is always 1, and for a point \((\cos(x), \sin(x))\) on the circumference, the equation \(\cos^2(x) + \sin^2(x) = 1^2\) naturally holds true.
The Pythagorean Identity is not only a foundational building block in trigonometry but is also used frequently for rewriting and simplifying expressions. It's a handy tool because it helps in expressing trigonometric functions in terms of each other, which is essential in solving trigonometric equations and proving identities.

Expression Simplification

Expression simplification involves reducing a trigonometric expression to a simpler form while maintaining its original value. It is crucial for solving trigonometric equations efficiently.
When simplifying expressions, knowing the basic identities like the Pythagorean Identity, and reciprocal identities such as \(\csc(x) = \frac{1}{\sin(x)}\) and \(\sec(x) = \frac{1}{\cos(x)}\), are invaluable.
For example, looking at the combination \(\sin^2(x) + \cos^2(x)\), with the Pythagorean Identity, we immediately know it simplifies to 1. This step can often make complex problems far easier to manage.
Simplification often involves adding terms to both sides of an equation, factorization, or rearrangement of terms to make solving more straightforward. When done correctly, simplified expressions provide a clearer insight into the problem without altering the expression's original meaning.

Trigonometric Manipulation

Trigonometric manipulation refers to the method of using various identities and algebraic techniques to rewrite trigonometric expressions. The aim is either to simplify them or to establish new identities.
In the example provided, we start with \(\sin^2(x) + \cos^2(x) = 1\) and manipulate it by introducing \(\tan^2(x)\). By adding \(\tan^2(x)\) to both sides and later using the identity \(\tan^2(x) = \sec^2(x) - 1\), we find a new and equivalent expression.
This manipulation process is key in trigonometry as it helps adapt or reformulate an equation to be fitted within a particular context or problem. Mastering trigonometric manipulation allows you to prove complex identities, solve equations that initially seem unrelated, and find solutions that otherwise might not be evident.
When learning trigonometric manipulation, it's important to practice applying different identities, and using them interchangeably, to grow more comfortable in transforming expressions accurately.