Problem 44 - Contemporary Precalculus

Question

In Exercises \(43-48,\) simplify the given expression. Assume that all denominators are nonzero and all quantities under radicals are nonnegative. $$\frac{\sec ^{2} t+2 \sec t+1}{\sec t}$$

Step-by-Step Solution

Answer

Question: Simplify the expression: $\frac{\sec ^{2} t + 2 \sec t +1}{\sec t}$, where all denominators are nonzero, and all quantities under radicals are nonnegative. Answer: $(\sec t + 1)\cos t(\sec t + 1)$

Step 1: Combine Like Terms

Notice that the expression has a secant squared and a secant term in the numerator. Let's group these terms together. $$\frac{\sec ^{2} t + 2 \sec t +1}{\sec t} = \frac{(\sec^2 t + 2\sec t) + 1}{\sec t}$$

Step 2: Factor the Numerator

We will now try to factor the terms inside the parentheses in the numerator: $$ (\sec^2 t + 2\sec t) = \sec t(\sec t + 2)$$ So the expression becomes: $$\frac{\sec t (\sec t + 2) + 1}{\sec t}$$

Step 3: Recognize a Perfect Square Pattern

Notice that the numerator could be written in the form of a perfect square if we add 1 inside the parentheses: $$\sec t (\sec t + 2) + 1 = (\sec t + 1)^2$$ Now the expression looks like: $$\frac{(\sec t + 1)^2}{\sec t}$$

Step 4: Simplify the Expression

Finally, we can simplify the expression, taking into account that all denominators are nonzero (thus \(\sec t \neq 0\)): $$\frac{(\sec t + 1)^2}{\sec t} = \frac{(\sec t + 1)(\sec t + 1)}{\sec t} = \frac{\sec t + 1}{\frac{1}{\sec t}}(\sec t + 1) = (\sec t + 1)\cos t(\sec t + 1)$$ Therefore, the simplified expression is: $$\boxed{(\sec t + 1)\cos t(\sec t + 1)}$$

Key Concepts

Trigonometric Identities

Understanding trigonometric identities is crucial for simplifying trigonometric expressions. These identities are equations that hold true for any value of the variable within their domains. They can be used to substitute one trigonometric function for another, making complex expressions more manageable. Some fundamental identities include the Pythagorean identities, the co-function identities, and the reciprocal identities.

For example, in trigonometric problems, knowing that \(\sec t\) is the reciprocal of \(\cos t\), which is written as \(\sec t = \frac{1}{\cos t}\), allows us to transition between functions effortlessly, leading to easier simplifications and solutions. As in the given exercise, understanding that \(\sec t\) is not zero will enable us to multiply or divide by \(\sec t\) safely, which are acts at the heart of simplifying trigonometric expressions.

Factoring Expressions

Factoring expressions is a powerful algebraic tool used to break down polynomials into simpler components known as factors. This technique is not limited to traditional polynomials but also extends to trigonometric expressions. By identifying common factors, we can simplify expressions and solve equations more efficiently.

In our exercise, the expression \(\sec^2 t + 2\sec t\) initially appears complicated. However, recognizing that both terms share a \(\sec t\) allows us to factor it out, resulting in \(\sec t(\sec t + 2)\). Factoring provides a pathway to further simplification and should be one of the first steps attempted when reducing the complexity of a trigonometric expression.

Perfect Square Trinomials

Perfect square trinomials are algebraic expressions formed by squaring a binomial. The standard form is \((a + b)^2 = a^2 + 2ab + b^2\). Recognizing these patterns in trigonometric contexts can lead to significant simplifications.

In the solution given, we observe this pattern emerge. The numerator \(\sec t (\sec t + 2) + 1\) resembles the expanded form of a binomial square, leading to the identification of \((\sec t + 1)^2\). Not only does this recognition reduce the complexity of the expression, but it also allows for a clearer path to the final simplified result, avoiding common pitfalls such as misapplication of the distributive property.

Precalculus Simplification Techniques

Precalculus simplification techniques encompass various strategies to condense and clarify mathematical expressions. This includes applying distributive properties, utilizing exponent rules, and canceling common factors. Sensibly combining these techniques allows students to approach complex expressions with confidence.

In our exercise, once we've factored the expression and recognized the perfect square, we simplify by canceling terms. Since we're given that all denominators are nonzero, we can safely divide by \(\sec t\), which corresponds to multiplying by \(\cos t\), its reciprocal. This transforms the expression into \((\sec t + 1)\cos t(\sec t + 1)\). Such techniques are critical for success in precalculus and beyond, laying a robust foundation for calculus concepts.