Problem 31 - Algebra 2

Question

Simplify each trigonometric expression. $$ \sec \theta \cos \theta-\cos ^{2} \theta $$

Step-by-Step Solution

Answer

The simplified version of the given trigonometric expression is \(\sin^2 \theta\).

Step 1: Rewrite Sec in terms of Cos

First step is to rewrite the secant function, \(\sec \theta\), in terms of cosine, as \(\frac{1}{\cos \theta}\). Therefore, the expression becomes \[\frac{1}{\cos \theta} \cdot \cos \theta - \cos^2 \theta\]

Step 2: Simplify the equation

Next, simplify \( \frac{1}{\cos \theta} \cdot \cos \theta\). The \(\cos \theta\) in the numerator and denominator cancels out, leaving you with \(1 - \cos^2 \theta\). Thus, the whole expression simplifies to \(1 - \cos^2 \theta\)

Step 3: Replace using Pythagorean identity

Finally, we can replace \(1 - \cos^2 \theta\) using the Pythagorean identity. The Pythagorean identity states that \(1 = \cos^2 \theta + \sin^2 \theta\), solving for \(\sin^2 \theta\) gives us \(\sin^2 \theta = 1 - \cos^2 \theta\). So the final expression simplifies to \(\sin^2 \theta\).

Key Concepts

Simplifying Trigonometric Expressions

Trigonometric expressions might look complicated, but by breaking them down into simpler parts, they become more manageable. The key is to identify terms or factors that can be rewritten using basic trigonometric identities or simplifications.
This involves recognizing relationships between trigonometric functions and transforming them to lesser or more convenient forms. For example:

• Changing secant (\(\sec \theta\)) into its cosine equivalent, \(\frac{1}{\cos \theta}\).
• Using the property that multiplying any expression by its reciprocal will result in 1, as seen when \(\frac{1}{\cos \theta} \cdot \cos \theta = 1\).

Applying these methods assists in canceling out terms and simplifying expressions to their core components, making math less intimidating and more intuitive.

Pythagorean Identity

The Pythagorean identity is a foundational trigonometric principle that helps in transforming and simplifying expressions. It states: \(1 = \cos^2 \theta + \sin^2 \theta\).
This identity demonstrates the relationship between sine and cosine, implying that the square of the sine plus the square of the cosine of any angle equals 1. This identity becomes essential when dealing with expressions such as \(1 - \cos^2 \theta\).
By rearranging the identity to solve for one trigonometric term, like \(\sin^2 \theta = 1 - \cos^2 \theta\), you create new avenues for simplification and synthesis in problems. Utilizing this identity simplifies trigonometric expressions by substituting complex terms with simpler equivalents, enabling easier handling of trigonometric problems.

Secant and Cosine Relationship

The secant and cosine relationship is a vital concept in trigonometry. These functions are directly related, as the secant function is defined as the reciprocal of the cosine function. Mathematically, it is expressed as \(\sec \theta = \frac{1}{\cos \theta}\).
Often in trigonometry, moving between these related functions allows for easier simplification of expressions. For instance, in the given expression, converting \(\sec \theta\) into \(\frac{1}{\cos \theta}\) was a crucial step that enabled further simplification when multiplied by \(\cos \theta\), effectively canceling out \(\cos \theta\) and simplifying the expression to more manageable terms.

• This relationship is not merely an academic concept but a practical tool in simplifying and solving complicated trigonometric formulas.
• It showcases the importance of understanding and utilizing function relationships in mathematics.

By grasping these transformations, trigonometric expressions become less daunting and more approachable.