Problem 25 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 21-26: Verify the identity by transforming the lefthand side into the right-hand side. $$ \frac{1}{\cos (-x)}-\tan (-x) \sin (-x)=\cos x $$

Step-by-Step Solution

Answer

The identity is verified: both sides simplify to \( \cos(x) \).

Step 1: Understand the Negative Angle Identity

First, recall the trigonometric identity for cosine of a negative angle: \( \cos(-x) = \cos(x) \). This means that the negative angle identity does not change the value of the cosine function.

Step 2: Substitute Cosine Identity

Replace \( \cos(-x) \) with \( \cos(x) \) in the left-hand side of the equation: \( \frac{1}{\cos(x)} - \tan(-x) \sin(-x) \).

Step 3: Apply Negative Angle Identities to Remaining Terms

Recall the identities for tangent and sine of negative angles: \( \tan(-x) = -\tan(x) \) and \( \sin(-x) = -\sin(x) \). Substitute these identities into the expression: \[ \frac{1}{\cos(x)} - (-\tan(x)) (-\sin(x)) \].

Step 4: Simplify the Expression

Simplify the expression using the identity substitutions: \[ \frac{1}{\cos(x)} - \tan(x) \sin(x) = \frac{1}{\cos(x)} - \frac{\sin(x)}{\cos(x)} \sin(x) \].

Step 5: Combine the Terms on a Common Denominator

Both terms have a common denominator of \( \cos(x) \). Combine the terms: \[ \frac{1 - \sin^2(x)}{\cos(x)} \].

Step 6: Recognize the Pythagorean Identity

Use the Pythagorean identity \( 1 - \sin^2(x) = \cos^2(x) \) to simplify further: \[ \frac{\cos^2(x)}{\cos(x)} \].

Step 7: Simplify to the Right-Hand Side

Simplify \( \frac{\cos^2(x)}{\cos(x)} \) by canceling \( \cos(x) \) from the numerator and the denominator: \( \cos(x) \). This matches the right-hand side of the equation.

Key Concepts

Negative Angle Identities

In trigonometry, negative angle identities are incredibly useful as they simplify expressions involving negative angles. When you have a negative angle, such as in the function \(-x\), the identity helps convert it into a positive angle. For cosine, the negative angle identity states that \(\cos(-x) = \cos(x)\), meaning cosine is an even function; back and forth across the y-axis mirrors it.
This retaining symmetry implies that the value remains unchanged for negative input.

• Cosine: \(\cos(-x) = \cos(x)\)
• Sine: \(\sin(-x) = -\sin(x)\)
• Tangent: \(\tan(-x) = -\tan(x)\)

These identities are vital in transforming equations, as they allow us to maintain consistent values while potentially flipping the sign of functions like sine and tangent, thereby simplifying the problem-solving process.

Pythagorean Identity

One of the cornerstone concepts in trigonometry is the Pythagorean identity. It is based on the Pythagorean theorem and states: \( \sin^2(x) + \cos^2(x) = 1 \). This identity helps to connect sine and cosine functions, allowing one to be expressed in terms of the other.
For example, if you know the value of \( \sin(x) \), you can calculate \( \cos(x) \) without directly measuring it, as long as the sign is appropriate for the angle's quadrant.

• Use case: Simplifying trigonometric expressions by substituting \( \cos^2(x) \) or \( \sin^2(x) \).
• Helps in finding one trigonometric value when the other is known.

This identity is powerful because it emerges in various applications, helping simplify expressions or verify other identities, just like in the given exercise where \( 1 - \sin^2(x) \) was substituted with \( \cos^2(x) \).

Angle Transformation

Angle transformations are essential when working with trigonometric expressions, particularly in verifying identities. Such transformations involve converting angles to more convenient forms, typically using known identities like the negative angle identities or angle sum and difference formulas.
In our exercise, the transformation involves turning negative angles into their positive equivalents, allowing more straightforward manipulation.

• Transforms complex expressions into simpler forms using known identities.
• Makes calculations generally more manageable when negative or large angles are involved.

Working with transformed angles can often be the first step in simplifying a problem or reaching the solution efficiently. It's about choosing the right transformations that align with the identities and simplify subsequent calculations.

Cosine Function

The cosine function is one of the primary functions in trigonometry, and it's crucial to understand its properties and how it behaves under different transformations. Cosine is an even function, which means that it remains unchanged when evaluated at a negative angle, as in \(\cos(-x) = \cos(x)\).
It starts its cycle with a value of 1 at 0 degrees, and its role in the unit circle is key: it represents the x-coordinate.

• Even function: Symmetrical about the y-axis.
• Periodicity: Completes a cycle every \(360^\circ\) or \(2\pi\).
• Occurs frequently in identities and transformations.

Understanding the properties of the cosine function and its identity relationships allow for effective problem solving, particularly in simplifying complex trigonometric expressions and verifying identities as seen in this exercise.