Problem 25 - Precalculus: Functions and Graphs

Question

Verify the identity by transforming the left hand side into the right-hand side. $$\frac{1}{\cos (-x)}-\tan (-x) \sin (-x)=\cos x$$

Step-by-Step Solution

Answer

The left side simplifies to \( \cos x \), confirming the identity.

Step 1: Simplify the Left Hand Side

The expression on the left hand side is \( \frac{1}{\cos(-x)} - \tan(-x) \sin(-x) \). First, recall the trigonometric identities: \( \cos(-x) = \cos x \) and \( \tan(-x) = -\tan x \). Substitute these into the expression to get \( \frac{1}{\cos x} + \tan x \sin(-x) \).

Step 2: Simplify Further Using Sine Identity

Since \( \sin(-x) = -\sin x \), substitute into the expression to get \( \frac{1}{\cos x} - \tan x \cdot (-\sin x) \), which simplifies to \( \frac{1}{\cos x} + \tan x \sin x \).

Step 3: Express \(\tan x \sin x\) in Terms of \(\cos x\) and \(\sin x\)

Knowing \( \tan x = \frac{\sin x}{\cos x} \), substitute this into the expression: \( \tan x \sin x = \frac{\sin x \cdot \sin x}{\cos x} = \frac{\sin^2 x}{\cos x} \). Thus, the whole expression is \( \frac{1}{\cos x} + \frac{\sin^2 x}{\cos x} \).

Step 4: Combine Fractions

Combine the fractions by adding the two terms: \( \frac{1 + \sin^2 x}{\cos x} \).

Step 5: Apply Pythagorean Identity

Use the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \) to replace \( 1 + \sin^2 x \) with \( \cos^2 x + \sin^2 x = 1 \). Therefore, the expression \( \frac{1 + \sin^2 x}{\cos x} \) becomes \( \frac{\cos^2 x + \sin^2 x}{\cos x} = \cos x \).

Step 6: Confirm the Identity

Thus, we have simplified the left hand side expression to \( \cos x \), which confirms that \( \frac{1}{\cos(-x)} - \tan(-x)\sin(-x) = \cos x \).

Key Concepts

Understanding the Cosine Function

The cosine function is a fundamental part of trigonometry. It is often abbreviated as "cos" and is one of the basic elementary trigonometric functions. The cosine of an angle in a right-angled triangle is defined as the ratio of the length of the adjacent side to the hypotenuse.

Here are some key properties of the cosine function:

• Periodic: The cosine function is periodic with a period of \(2\pi\). This means that the function repeats its values every \(2\pi\) radians.
• Even Function: The cosine function is an even function, indicating that \(\cos(-x) = \cos(x)\) for any angle \(x\). This property was used in the solution to transform \(\frac{1}{\cos(-x)}\) into \(\frac{1}{\cos x}\).
• Range and Domain: The range of the cosine function is from -1 to 1, and its domain is all real numbers.

Understanding the cosine function's behavior helps in analyzing identities and simplifying trigonometric expressions. It is pivotal in the formulaic manipulations seen in the given solution.

Exploring the Tangent Function

The tangent function, often represented as "tan," is another crucial trigonometric function. It is defined as the ratio of the sine of an angle to the cosine of the same angle. That is, \( \tan(x) = \frac{\sin(x)}{\cos(x)} \).

Some significant characteristics of the tangent function:

• Odd Function: The tangent function is an odd function, meaning \(\tan(-x) = -\tan(x)\). This is applied when simplifying trigonometric identities, as seen in the exercise where \(\tan(-x)\) is transformed into \(-\tan(x)\).
• Periodicity: Unlike sine and cosine, the tangent function has a period of \(\pi\), reflecting its repeating cycle.
• No Bound Limit: The range of the tangent function is all real numbers, with vertical asymptotes where the cosine function equals zero.
• Combination: The expression \(\tan x \sin x = \frac{\sin^2 x}{\cos x}\) illustrates how the tangent function can be rewritten, as shown in the step-by-step solution.

Grasping the characteristics of the tangent function is essential when verifying or manipulating trigonometric identities, allowing you to transform complex expressions into simpler equivalent ones.

Delving into the Sine Function

The sine function is one of the basic trigonometric functions, symbolized as "sin." In the context of a right triangle, it is defined as the ratio of the length of the opposite side to the hypotenuse for a given angle.

Key features of the sine function include:

• Odd Functionality: The sine function is an odd function so \(\sin(-x) = -\sin(x)\). This was utilized in the exercise to change \(\sin(-x)\) into \(-\sin(x)\), helping simplify the expression.
• Periodicity: The sine function, like cosine, has a periodicity of \(2\pi\), repeating its pattern every \(2\pi\) radians.
• Range of Values: The range of \( \sin(x) \) is from -1 to 1, making it bounded within these limits.
• Pythagorean Identity Integration: The sine function is part of the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\), crucial in simplifying the given identity from the exercise.

Understanding the sine function, and its role in identities, is fundamental for solving trigonometric equations and for transforming and verifying trigonometric expressions as seen in this exercise.