Problem 144 - Calculus Volume 1

Question

Simplify each expression by writing it in terms of sines and cosines, then simplify. The final answer does not have to be in terms of sine and cosine only. \(\sin x(\csc x-\sin x)\)

Step-by-Step Solution

Answer

The expression simplifies to \( \cos^2 x \).

Step 1: Identify and Write Expressions in Terms of Sine and Cosine

The original expression is \( \sin x (\csc x - \sin x) \). We can rewrite \( \csc x \) in terms of sine as \( \csc x = \frac{1}{\sin x} \). Substituting, the expression becomes \( \sin x \left( \frac{1}{\sin x} - \sin x \right) \).

Step 2: Simplify Using Basic Arithmetic

The expression now is: \( \sin x \left( \frac{1}{\sin x} - \sin x \right) = \sin x \times \frac{1}{\sin x} - \sin^2 x \). Simplify by performing the multiplication: \( 1 - \sin^2 x \).

Step 3: Recognize and Use Trigonometric Identities

Using the Pythagorean identity, we recognize that \( 1 - \sin^2 x \) is equivalent to \( \cos^2 x \). Hence, the expression simplifies to \( \cos^2 x \).

Key Concepts

Sine and Cosine Conversion

In trigonometry, converting expressions into sines and cosines can make complex problems much easier to work with. This is because sine (\(\sin x\)) and cosine (\(\cos x\)) are often foundational elements in trigonometric identities. For the expression \(\csc x\) (cosecant of \(x\)), it can be represented in terms of \(\sin x\) as \(\csc x = \frac{1}{\sin x}\). Changing your given expression into these terms is often the first step towards simplification.
When you have an expression like \(\sin x (\csc x - \sin x)\), converting \(\csc x\) into \(\frac{1}{\sin x}\) allows us to decompose the expression entirely in terms of sine. This sets up for easier cancellation and further simplification. Converting to a single trigonometric form brings clarity and consistency to the steps that follow.

Simplifying Expressions

Simplifying trigonometric expressions often involves basic algebraic principles, such as distribution and combining like terms. For the expression \(\sin x \left( \frac{1}{\sin x} - \sin x \right)\), we'd first distribute \(\sin x\) across the terms inside the parentheses. This simple multiplication step leads us to \(\sin x \times \frac{1}{\sin x} - \sin^2 x\).
By simplifying \(\sin x \times \frac{1}{\sin x}\), we reach \(1\) because the sine terms cancel each other out. Thus, the expression condenses to \(1 - \sin^2 x\), bringing us closer to our simplified form. Understanding and executing these arithmetic operations efficiently is key to mastering trigonometric problems.

Pythagorean Identity

Trigonometry is full of useful identities, and one of the most valuable is the Pythagorean identity. This identity relates the square of sine and cosine: \(\sin^2 x + \cos^2 x = 1\). Recognizing this, whenever you see something like \(1 - \sin^2 x\), you can substitute it with \(\cos^2 x\). This transformation is the final piece when simplifying the given expression.
In the expression \(1 - \sin^2 x\), knowing that it equals \(\cos^2 x\) via the Pythagorean identity simplifies our work tremendously. This shows the deep connection between sine and cosine. Knowing the Pythagorean identity provides a direct route from a somewhat complicated expression to its simplest form, allowing us to finalize our solution to arrive at \(\cos^2 x\) efficiently. Mastery of such identities allows one to solve and restructure trigonometric expressions swiftly.