Problem 1 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-50: Verify the identity. $$ \csc \theta-\sin \theta=\cot \theta \cos \theta $$

Step-by-Step Solution

Answer

The identity is verified: both sides simplify to \( \frac{\cos^2 \theta}{\sin \theta} \).

Step 1: Express in terms of Sine and Cosine

We start by expressing all trigonometric identity terms using sine and cosine.- The cosecant function can be expressed as: \( \csc \theta = \frac{1}{\sin \theta} \)- The cotangent function can be expressed as: \( \cot \theta = \frac{\cos \theta}{\sin \theta} \)So the left-hand side becomes: \( \frac{1}{\sin \theta} - \sin \theta \) and the right-hand side is \( \frac{\cos^2 \theta}{\sin \theta} \)

Step 2: Combine the Left-Hand Side

Now we find a common denominator for the left-hand side.\[ \frac{1}{\sin \theta} - \sin \theta = \frac{1 - \sin^2 \theta}{\sin \theta} \]

Step 3: Simplify Using Pythagorean Identity

Use the Pythagorean identity: \( 1 - \sin^2 \theta = \cos^2 \theta \).Substitute this into the expression:\[ \frac{1}{\sin \theta} - \sin \theta = \frac{\cos^2 \theta}{\sin \theta} \]

Step 4: Equate Both Sides

Now that both sides of the equation are the same: \( \frac{\cos^2 \theta}{\sin \theta} \).The identity \( \csc \theta - \sin \theta = \cot \theta \cos \theta \) is verified.

Key Concepts

Cosecant Function

The cosecant function is one of the lesser-known trigonometric functions, but it plays a crucial role in many mathematical problems. It is often abbreviated as "csc." The cosecant of an angle \( \theta \) is the reciprocal of the sine function. This means that it is defined as:

• \( \csc \theta = \frac{1}{\sin \theta} \)

This simple relationship transforms sine-based expressions into its reciprocal form. It is important for solving trigonometric identities or equations where direct division could simplify the expressions.
When working with the cosecant function, always remember it is undefined for angles where the sine is zero, since division by zero is not possible. This typically happens at angles where \( \theta = n\pi \), where \( n \) is an integer.

Cotangent Function

The cotangent function is another important trigonometric function, typically abbreviated as "cot." It is related closely to both the sine and cosine functions. The definition of cotangent is:

• \( \cot \theta = \frac{\cos \theta}{\sin \theta} \)

This function is essentially the reciprocal of the tangent function, but in terms of sine and cosine, making it extremely useful in simplifying trigonometric identities.
However, just like the cosecant function, cotangent also becomes undefined when \( \theta = n\pi \), because the sine of these angles is zero, leading to division by zero scenarios. Understanding these restrictions helps in preventing errors when verifying or manipulating trigonometric identities.

Pythagorean Identity

The Pythagorean identity is a fundamental relation in trigonometry that you will frequently use to simplify expressions and verify identities. This identity is directly derived from the Pythagorean theorem and involves sine and cosine functions:

• \( \sin^2 \theta + \cos^2 \theta = 1 \)

An essential rearrangement of this formula is:

• \( 1 - \sin^2 \theta = \cos^2 \theta \)

This rearrangement helps in solving identities by transforming terms from sine to cosine, or vice versa. In our given exercise, using this identity transformed the left-hand expression into the same form as the right-hand expression, validating the trigonometric identity. Memorizing and understanding these basic identities is key to tackling more complex trigonometric problems.

Sine and Cosine Expressions

Sine and cosine are the most fundamental trigonometric functions. They are the building blocks for defining other trigonometric expressions and identities.

• The sine function, denoted as \( \sin \theta \), relates the ratio of the opposite side to the hypotenuse in a right triangle.
• The cosine function, denoted as \( \cos \theta \), relates the ratio of the adjacent side to the hypotenuse.

Combined, they form the basis of understanding and simplifying more complex trigonometric expressions. In many cases, like the exercise provided, trigonometric expressions are simplified by rewriting everything in terms of sine and cosine.
This strategy aids in simplifying the terms and making algebraic manipulations easier, eventually leading you to verify given identities, or solve trigonometric equations with clarity.