Problem 29 - Algebra 2

Question

Simplify each trigonometric expression. $$ \cos \theta\left(1+\tan ^{2} \theta\right) $$

Step-by-Step Solution

Answer

The simplification of the trigonometric expression \(\cos \theta(1+\tan ^{2} \theta)\) is \(1/\cos \theta\).

Step 1: Apply the Pythagorean Identity

We know that \(1 + \tan ^{2} \theta = \sec ^{2} \theta\). Substituting this into our expression we get \(\cos \theta * \sec ^{2} \theta\).

Step 2: Recall the relationship between cosine and secant

Secant is the reciprocal of cosine, \(\sec \theta = 1/\cos \theta\). Rewriting our expression using this relationship, we get \(\cos \theta * (1/\cos \theta) ^ {2}\).

Step 3: Simplify the expression

Given the expression \(\cos \theta * 1/(\cos^2 \theta)\), we see that the numerator and denominator cancel out to give us \(1/\cos \theta\).

Key Concepts

Simplifying Trigonometric Expressions

Trigonometric expressions often look complex, but simplifying them makes solving related problems easier. By breaking them down, we can transform intricate equations into understandable forms. For example, expressions like \( \cos \theta\left(1+\tan ^{2} \theta\right) \) might seem difficult initially. But using known identities and relationships, we can turn them into simpler expressions.

Simplification is all about recognizing parts of the expression that can be substituted or rewritten. Consider how we use identities like the Pythagorean Identity to replace parts of expressions. This approach transforms a tangled trigonometric equation into something elegant and manageable. Always look for identities and algebraic manipulations in your tools when simplifying.

Pythagorean Identity

The Pythagorean Identity is one of the central connections in trigonometry. It shows the relationship between \( \sin \theta \), \( \cos \theta \), and \( \tan \theta \). The primary form of this identity is:

• \( \sin^2 \theta + \cos^2 \theta = 1 \)

From this primary form, we can derive others, like:

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

This last version is particularly useful in our example. It allows us to replace \( \tan^2 \theta \) with \( \sec^2 \theta - 1 \), simplifying calculations. By recognizing these identities, you can disassemble and piece together expressions in more manageable forms.

Cosine and Secant Relationship

Cosine and secant are closely tied through the reciprocal nature of trigonometric functions. Specifically, secant is the reciprocal of cosine:

• \( \sec \theta = \frac{1}{\cos \theta} \)

This relationship means that any time you see \( \sec \theta \), you can think of it as \( \frac{1}{\cos \theta} \). This is crucial when simplifying expressions. In our solution, replacing \( \sec^2 \theta \) converts the whole expression to something involving only cosine.

Understanding these relationships helps decode the complexity of compound expressions. Such insights make managing and solving trigonometric problems more straightforward.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions include secant, cosecant, and cotangent. They are defined in terms of sine, cosine, and tangent:

• \( \sec \theta = \frac{1}{\cos \theta} \)
• \( \csc \theta = \frac{1}{\sin \theta} \)
• \( \cot \theta = \frac{1}{\tan \theta} \)

These reciprocal functions are handy shortcuts in trigonometric identities and equations. When working with trigonometric expressions, substituting in their reciprocal forms can simplify and clarify complex situations.

In the example expression \( \cos \theta \times \sec^2 \theta \), understanding that \( \sec \theta \) equals \( \frac{1}{\cos \theta} \) allowed us to simplify it to \( \frac{1}{\cos \theta} \). This kind of transformation, using reciprocals, turns challenging expressions into simpler forms that are easier to interpret and solve.