Problem 104 - 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

\( \frac{1}{\tan x} + \tan x = \frac{2}{\sin(2x)} \).

Step 1: Start with a Trigonometric Expression

Begin with a given trigonometric expression, such as \( rac{1}{ an x} + an x \). The task is to simplify or rewrite this expression to form an identity.

Step 2: Rewrite the Expression Using Identities

Rewrite the expression by recalling the identity \( \tan x = \frac{\sin x}{\cos x} \). This changes \( \frac{1}{\tan x} + \tan x \) to \( \frac{1}{ \frac{\sin x}{\cos x}} + \frac{\sin x}{\cos x} \).

Step 3: Simplify the Expression

Simplify \( \frac{1}{ \frac{\sin x}{\cos x}} \) to \( \frac{\cos x}{\sin x} \). Combine the terms: \( \frac{\cos x}{\sin x} + \frac{\sin x}{\cos x} \).

Step 4: Find a Common Denominator

To add the fractions, find a common denominator: \( \frac{\cos^2 x + \sin^2 x}{\sin x \cos x} \).

Step 5: Apply the Pythagorean Identity

Identify \( \cos^2 x + \sin^2 x \) as the Pythagorean identity, which equals 1. Substitute this back into the equation: \( \frac{1}{\sin x \cos x} \).

Step 6: Simplify to Complete the Identity

Recognize that \( \frac{1}{\sin x \cos x} \) is equivalent to \( \frac{2}{2 \sin x \cos x} = \frac{2}{\sin(2x)} \). Thus, the identity is: \( \frac{1}{\tan x} + \tan x = \frac{2}{\sin(2x)} \).

Key Concepts

Trigonometric Expressions

Trigonometric expressions involve functions such as sine, cosine, tangent, and their reciprocal functions like secant, cosecant, and cotangent. These expressions are used to describe relationships between angles in right triangles. In algebraic terms, a trigonometric expression can be a combination of these functions. Such expressions are often used in trigonometry to define specific angles and solve problems regarding triangles and circles.

To manipulate trigonometric expressions, you can use arithmetic operations like addition and subtraction. Often, the goal is to simplify or rewrite the expression. By simplifying, you reduce complexity, making it easier to work with or solve an equation.

Here are some common trigonometric expressions you might encounter:

• Simple forms like \( \sin x \) or \( \cos x \)
• Fractions, such as \( \frac{1}{\tan x} \)
• Combinations like \( \tan x + \sin x \)
• Products, e.g., \( \sin x \cos x \)

Understanding and manipulating these expressions is crucial for solving more complex problems.

Rewriting or simplifying a trigonometric expression can lead to the creation of identities, which hold true for all values of the angle.

Pythagorean Identity

The Pythagorean identity is a fundamental concept in trigonometry, expressing the intrinsic relationship between sine and cosine functions. The formula is given by \( \cos^2 x + \sin^2 x = 1 \). This identity arises from the Pythagorean theorem applied in the context of the unit circle.

It states that for any angle, the square of the cosine of the angle plus the square of the sine of the angle always sums up to one. This is valid for all angles and is used to simplify expressions and prove other identities.

Lets illustrate with an example:

• Given \( \cos^2 x + \sin^2 x \) in an expression, you can substitute it directly with 1.
• This can greatly simplify the expression, as seen in the step-by-step solution, where \( \cos^2 x + \sin^2 x \) was replaced by 1, facilitating the formulation of a new identity.

Understanding this identity allows you to manipulate and simplify trigonometric expressions efficiently.

This identity is a handy tool for trigonometric simplifications and is often used as a stepping stone to derive other such identities.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions involves reducing them to simpler or more manageable forms. The process can include combining like terms, factoring, or substituting known identities.

In the provided step-by-step solution, we simplified an expression involving \( \frac{1}{\tan x} + \tan x \), which at first may seem complex.

Simplification techniques include:

• Rewriting terms using basic identities, like changing \( \tan x \) to \( \frac{\sin x}{\cos x} \).
• Combining fractions by finding a common denominator.
• Using identities like the Pythagorean identity to substitute and reduce terms.
• Re-expressing complex fractions in simpler forms.

These techniques help in transforming a complex expression into a simpler equivalent, making problem-solving more straightforward.

Ultimately, through simplification, expressions become easier to work with, lending themselves to forming meaningful identities or solutions.