Problem 37 - Contemporary Precalculus

Question

In Exercises \(37-42\), factor and simplify the given expression. $$\sec t \csc t-\csc ^{2} t$$

Step-by-Step Solution

Answer

Answer: \(\frac{1 - \cos t}{\cos t \cdot \sin t}\).

Step 1: Rewrite the expression using basic trigonometric identities

We can rewrite the expression using the following basic trigonometric identities: \(\sec t = \frac{1}{\cos t}\), \(\csc t = \frac{1}{\sin t}\). So, the given expression becomes: $$\frac{1}{\cos t} \cdot \frac{1}{\sin t} - \left(\frac{1}{\sin t}\right)^2$$

Step 2: Simplify the expression

Now, let's simplify the expression: $$\frac{1}{\cos t \cdot \sin t} - \frac{1}{\sin^2 t}$$

Step 3: Find a common denominator and combine the terms

In order to combine these two terms, we need to find a common denominator. In this case, the common denominator is \(\cos t \cdot \sin t\). We will rewrite each term with this common denominator and then combine them: $$\frac{1}{\cos t \cdot \sin t} - \frac{\cos t}{\cos t \cdot \sin t}$$ $$= \frac{1 - \cos t}{\cos t \cdot \sin t}$$

Step 4: Factor the expression

We can now factor the expression as follows: $$= \frac{(1 - \cos t)}{\cos t \cdot \sin t}$$ So, the factored and simplified expression for the given expression is: $$\frac{1 - \cos t}{\cos t \cdot \sin t}$$

Key Concepts

Secant

The secant function, often abbreviated as "sec," is a less commonly used trigonometric function but equally important in simplifying expressions and solving equations. It's defined as the reciprocal of the cosine function. This means that \(\sec t = \frac{1}{\cos t}\).
Understanding the secant function's relationship with cosine is crucial when working with trigonometric identities since it provides a different perspective to analyze angles and sides in a right triangle or unit circle context.
When you see \(\sec t\) in an expression, it might be helpful to consider it in terms of cosine\(^{-1}\), as this perspective can be valuable in simplifying complex expressions.

• Secant links directly to angles derived from the unit circle, where the function value represents the ratio of the line segment through the origin that is tangent to the circle at the angle \(t\).


• It frequently appears in oscillatory motion problems, wave equations, and other fields requiring periodic functions.

Establishing these reciprocal relationships paves the way for easily factoring expressions, as seen in exercises like the one you were working on.

Cosecant

Cosecant, abbreviated as ''csc,'' is another less common yet valuable trigonometric function. It's the reciprocal of the sine function, expressed as \(\csc t = \frac{1}{\sin t}\). Just like secant connects to cosine, the cosecant relates back to sine.
Unlike cosine and secant that use the x-coordinate in the unit circle, sine and cosecant work with the y-coordinate. Therefore, if an angle t is known, \(1/sin(t)\) gives the cosecant value, which is particularly useful when analyzing waves or alternating signals.
When facing trigonometric expressions involving \(\csc t\), remember:

• It's vital to use the identity to convert between sine and cosecant where necessary.


• Cosecant is significant in problems involving angles of elevation or calculations requiring the length of hypotenuses.

Transforming expressions from cosecant to sine terms, as shown in the original problem, helps simplify complex expressions and apply identities for easier calculations.

Simplifying Trigonometric Expressions

In mathematics, simplifying trigonometric expressions is crucial for tackling complex equations and making calculations more manageable. Simplification often involves converting higher functions into terms of sine, cosine, secant, or cosecant, as demonstrated in the original exercise.
The importance of simplifying trigonometric expressions includes:

• Combining multiple trigonometric identities into a unified expression.


• Reducing complexity for easier graphical interpretation or numerical computation.


• In exercises like the one given, reworking expressions with common denominators and recognizing factoring possibilities as major simplification techniques.

The process often starts by replacing complex trigonometric identities with basic ones, such as converting \(\sec t\) into \(\frac{1}{\cos t}\) or \(\csc t\) into \(\frac{1}{\sin t}\). These transformations allow the combination of terms with like denominators to facilitate factoring, significantly simplifying calculations.