Question

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

Short answer

The simplified expression is \( a^2 \sec^2 \theta \).

Step-by-step solution

Identify the Expression

The given expression is \( a^2 + a^2 \tan^2 \theta \).

Factor Out Common Terms

Notice that both terms in the expression have a common factor of \( a^2 \). By factoring it out, we rewrite the expression as \[ a^2 (1 + \tan^2 \theta) \].

Use Trigonometric Identity

Recall the identity \( 1 + \tan^2 \theta = \sec^2 \theta \). Replace \( 1 + \tan^2 \theta \) in the expression with \( \sec^2 \theta \). The expression becomes: \[ a^2 \sec^2 \theta \].

Key concepts

Trigonometric Identities

Trigonometric identities are equations that relate different trigonometric functions to one another. They provide a handy way to simplify complex expressions using known relationships. One widely used identity is the Pythagorean identity.This identity tells us that for any angle \( \theta \), the following is true:

• \( 1 + \tan^2 \theta = \sec^2 \theta \)

Being familiar with identities like these helps immensely in simplifying trigonometric expressions. In the given problem, recognizing the identity \( 1 + \tan^2 \theta = \sec^2 \theta \) was key to reducing the expression to a simpler form. In essence, knowing these identities allows you to replace parts of an expression with simpler equivalents, making calculations easier to handle.

Factoring

Factoring is a fundamental technique in algebra used to break down expressions into simpler components, usually by pulling out common terms. This process transforms complex expressions into more manageable pieces, which can then be simplified further.In our exercise, the expression \( a^2 + a^2 \tan^2 \theta \) contains the common factor \( a^2 \). By factoring out \( a^2 \), the expression becomes:

• \( a^2 (1 + \tan^2 \theta) \)

Factoring simplifies the process of applying trigonometric identities. Once factored, the expression becomes easier to read and manipulate. It's akin to decluttering a room; by organizing the pieces, you can clearly see how everything fits together and simplify your work more efficiently.

Trigonometric Expressions

Trigonometric expressions are mathematical phrases that involve trigonometric functions like sine, cosine, tangent, and their reciprocals (cosecant, secant, cotangent). Simplifying these expressions involves using trigonometric identities and algebraic techniques like factoring.The aim is to express them in the simplest form possible or in terms of a single trigonometric function. This makes them easier to interpret or solve in further equations or calculations. For example, in the given problem:

• The original expression was \( a^2 + a^2 \tan^2 \theta \)
• After applying a trigonometric identity and factoring, it simplified to \( a^2 \sec^2 \theta \)

Here, a complex expression was efficiently rewritten using a single known trigonometric function, \( \sec \theta \). Simplifying in such a manner helps make advanced calculations more straightforward and minimizes computational errors.