Problem 34 - Contemporary Precalculus

Question

In Exercises \(30-36,\) perform the indicated operations, then simplify your answers by using appropriate definitions and identities. $$(\sin t-\csc t)^{2}$$

Step-by-Step Solution

Answer

Question: Simplify the trigonometric expression \((\sin t - \csc t)^2\). Answer: \(\frac{(\cos^2 t)^2}{1-\cos^2 t}\)

Step 1: Write the Expression Using Definitions

Recall that \(\csc t = \frac{1}{\sin t}\). Rewrite the given expression using this definition: $$(\sin t - \frac{1}{\sin t})^2$$

Step 2: Simplify the Expression Using Common Denominator

Find a common denominator to combine the terms inside the parentheses. The common denominator for \(\sin t\) and \(\frac{1}{\sin t}\) is \(\sin t\). Write both terms with the common denominator. $$\left(\frac{\sin^2 t - 1}{\sin t}\right)^2$$

Step 3: Apply Trigonometric Identity for \(\sin^2 t\)

Use the Pythagorean trigonometric identity, \(\sin^2 t + \cos^2 t = 1\), to replace \(\sin^2 t\) in the expression. $$\left(\frac{(1-\cos^2 t) - 1}{\sin t}\right)^2$$

Step 4: Simplify and Find the Square

Simplify the expression inside the parentheses and then find the square of the resulting expression. $$\left(\frac{-\cos^2 t}{\sin t}\right)^2$$ $$= \frac{(\cos^2 t)^2}{(\sin t)^2}$$

Step 5: Apply Trigonometric Identity for \(\sin^2 t\) Again

Use the Pythagorean trigonometric identity, \(\sin^2 t + \cos^2 t = 1\), to replace \(\sin^2 t\) in the expression. $$= \frac{(\cos^2 t)^2}{(1-\cos^2 t)}$$ So, the simplified expression is: $$\frac{(\cos^2 t)^2}{1-\cos^2 t}$$

Key Concepts

Simplifying Expressions

Simplifying expressions involves reducing them to their most basic and concise form, making them easier to understand and evaluate. This often requires using mathematical identities and definitions to substitute and manipulate parts of the expression. In our given trigonometric expression \((\sin t - \csc t)^2\), simplification is crucial. We begin by remembering that the cosecant function \(\csc t\) is the reciprocal of the sine function, specifically \(\csc t = \frac{1}{\sin t}\). With this substitution, we rewrite the expression as \((\sin t - \frac{1}{\sin t})^2\). To simplify further, we must combine terms into a single fraction by obtaining a common denominator. This sometimes involves additional steps, such as adopting common denominators, expanding terms, and factoring where possible. Throughout the simplification process, it's essential to work methodically to prevent errors and ensure accuracy. Using known trigonometric identities efficiently can greatly simplify complex expressions, which leads us to the introduction of Pythagorean Identities in the next concept.

Pythagorean Identity

The Pythagorean Identity is a fundamental concept in trigonometry that connects the square of sine and cosine. It states that \(\sin^2 t + \cos^2 t = 1\). This identity is extremely useful for simplifying trigonometric expressions, especially when we need to express everything in terms of one trigonometric function. In our challenge, when encountering \(\sin^2 t\) in \(\frac{\sin^2 t - 1}{\sin t}\) during simplification, we can use the Pythagorean Identity to replace \(\sin^2 t\) with \(1 - \cos^2 t\). By substituting \(\sin^2 t\) this way, the expression becomes easier to handle: \(\left(\frac{1 - \cos^2 t - 1}{\sin t}\right)^2\) simplifies to \(\left(\frac{-\cos^2 t}{\sin t}\right)^2\). The application of this identity not only simplifies the expression but also helps in accurately evaluating or reducing complex trigonometric functions. It's a staple identity especially useful when dealing with right-angle triangles and can be leveraged across various trigonometric problems.

Trigonometric Operations

Trigonometric operations involve the main functions: sine, cosine, tangent, and their reciprocals – cosecant, secant, and cotangent. These operations are used to manipulate expressions and solve equations that involve trigonometric functions. They need specific trigonometric identities and algebraic strategies for simplification. In calculations like \((\sin t - \csc t)^2\), operations such as finding squares, subtracting functions, and simplifying fractions are crucial. Using the knowledge of the relationships between trigonometric functions is key. For instance, understanding that \(\csc t = \frac{1}{\sin t}\) allows us to transform and simplify expressions. Moreover, by applying identities like the Pythagorean Identity during these operations, you can manipulate the equations more fluidly and end up with a simpler and more manageable formula, such as \(\frac{(\cos^2 t)^2}{1-\cos^2 t}\). This makes it easier to apply further mathematical operations that might be required in solving or analyzing more complicated trigonometric problems.