Problem 26 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 21-26: Verify the identity by transforming the lefthand side into the right-hand side. $$ \cot (-x) \cos (-x)+\sin (-x)=-\csc x $$

Step-by-Step Solution

Answer

The identity is verified as both sides equal \(-\csc(x)\).

Step 1: Recognize the Even and Odd Functions

First, recall that the trigonometric functions have properties of being even or odd. An even function satisfies \( f(-x) = f(x) \), and an odd function satisfies \( f(-x) = -f(x) \). The cosine function \( \cos(x) \) is even, while the sine function \( \sin(x) \), and \( \cot(x) \) are odd. This means \( \cos(-x) = \cos(x) \), \( \sin(-x) = -\sin(x) \), and \( \cot(-x) = -\cot(x) \).

Step 2: Apply Properties to Left-hand Side

Substitute these properties into the left-hand side of the equation. The expression \( \cot(-x) \cos(-x) + \sin(-x) \) becomes \( -\cot(x) \cos(x) - \sin(x) \).

Step 3: Substitute \( \cot(x) \) and \( \csc(x) \) Identities

Recall that \( \cot(x) = \frac{\cos(x)}{\sin(x)} \) and \( \csc(x) = \frac{1}{\sin(x)} \). Substitute these into the equation, giving you \( -(\frac{\cos(x)}{\sin(x)}) \cos(x) - \sin(x) \).

Step 4: Simplify the Expression

Distribute the multiplication on the left-hand side: \( -\frac{\cos^2(x)}{\sin(x)} - \sin(x) \). We can combine these terms by writing -\( \sin(x) \) as -\( \frac{\sin^2(x)}{\sin(x)} \).

Step 5: Add the Fractions

Combine the fractions over a common denominator. Use \( \sin(x) \) as the common denominator: \( \frac{-\cos^2(x) - \sin^2(x)}{\sin(x)} \).

Step 6: Apply Pythagorean Identity

Recall the Pythagorean identity \( \cos^2(x) + \sin^2(x) = 1 \). Therefore, \( -\cos^2(x) - \sin^2(x) = -1 \). Substitute this into the equation: \( \frac{-1}{\sin(x)} \).

Step 7: Simplify to Right-hand Side

Recognize that \( \frac{-1}{\sin(x)} \) is just \( -\csc(x) \), which matches the right-hand side of the original identity.

Key Concepts

even and odd functions

Trigonometric functions have unique properties called even and odd characteristics that can simplify expressions and identities.
An **even function** has the property that the function evaluated at negative a value is the same as the function evaluated at the positive of that value. Mathematically, this is written as \( f(-x) = f(x) \). For trigonometric functions, the cosine function \( \cos(x) \) is even. This means that if you plug in \(-x\), the answer will be the same as if you simply used \(x\): \( \cos(-x) = \cos(x) \).
On the other hand, an **odd function** satisfies \( f(-x) = -f(x) \). This means the function's value at a negative input is the negative of the function's value at the positive input. For trigonometric functions, sine \( \sin(x) \) and cotangent \( \cot(x) \) are odd. Therefore, \( \sin(-x) = -\sin(x) \) and \( \cot(-x) = -\cot(x) \).
Understanding these properties helps to transform expressions. In the exercise, recognizing that \( \cot(-x) \) and \( \sin(-x) \) are odd and \( \cos(-x) \) is even, allows straightforward substitution into the left-hand side of the equation.

cotangent and cosecant identities

To work with trigonometric identities, it's essential to know the basic identities for different trigonometric functions. Two important identities are for **cotangent** and **cosecant**.
The **cotangent** identity is defined as the ratio of cosine to sine. Mathematically, this is expressed as \( \cot(x) = \frac{\cos(x)}{\sin(x)} \). This identity is useful for transforming complex trigonometric expressions into simpler forms by breaking them down into basic sine and cosine terms.
The **cosecant** identity is the reciprocal of the sine function. It is given by \( \csc(x) = \frac{1}{\sin(x)} \). Using this identity, you can often simplify expressions involving cosecant by rewriting them in terms of sine.
In this exercise, these identities were used to rewrite \( \cot(x) \) and \( \csc(x) \) in a form that made it possible to simplify and verify the given identity. By substituting \( \cot(x) = \frac{\cos(x)}{\sin(x)} \) and expressing other terms in terms of sine, complex fractions can be combined and simplified effectively.

Pythagorean identity

One of the most fundamental identities in trigonometry is the **Pythagorean identity**. It states that for any angle \( x \), the square of the cosine plus the square of the sine is always equal to one. The formula is \( \cos^2(x) + \sin^2(x) = 1 \).
This identity is incredibly useful in simplifying trigonometric expressions and in verifying more complicated identities. By rearranging the Pythagorean identity, you can solve for different terms:

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

In this exercise, the identity was used to convert complex expressions into simpler ones. Specifically, by recognizing that \( -\cos^2(x) - \sin^2(x) = -1 \), you use this equality to simplify how the fractions are combined. This simplification directly leads to the expression's transformation into \( -\frac{1}{\sin(x)} \), verifying the original problem's identity of \( -\csc(x) \).
Mastering the Pythagorean identity is crucial for tackling a wide range of trigonometric problems.