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Chapter 5: Problem 25

Verifying a Trigonometric Identity Verify the identity. $$\sec x-\cos x=\sin x \tan x$$

Short Answer

Expert verified
The trigonometric identity \( \sec x-\cos x=\sin x \tan x \) is verified through a series of steps that require understanding of fundamental trigonometric identities like \( \sec x = \frac{1}{\cos x} \) and \( \tan x = \frac{\sin x}{\cos x} \). By simplifying these parts accordingly, we find that the identity holds true.

Step by step solution

01

Simplify the left side of the equation

On the left side, remember that the secant of x or \( \sec x \) is equal to \( \frac{1}{\cos x} \). So, replace \( \sec x \) with \( \frac{1}{\cos x} \). It is now \( \frac{1}{\cos x} -\cos x \).
02

Simplify the right side of the equation

On the right side, remember that the tangent of x or \( \tan x \) is equal to \( \frac{\sin x}{\cos x} \). So, replace \( \tan x \) with \( \frac{\sin x}{\cos x} \). It is now \( \sin x \cdot \frac{\sin x}{\cos x} \).
03

Combine the simplified left and right side

When the left and right side of the equation are combined, it forms \( \frac{1}{\cos x} -\cos x = \sin x \cdot \frac{\sin x}{\cos x} \). Next, simplify further by combining like terms.
04

Further simplifying the equation

We can now multiply the right side by \( \cos x \) and the left side by \( \cos x \) to cancel out the \( \cos x \) in the denominators of both sides. After that, the equation becomes \( 1 - \cos^2 x = \sin^2 x \). This equation is true because of Pythagorean Identity that \( \sin^2 x + \cos^2 x = 1 \). Hence, the identity is verified.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Secant Function
The secant function, denoted as \( \sec x \), is a trigonometric function that is the reciprocal of the cosine function. In simpler terms, the secant of an angle \( x \) is defined as \( \sec x = \frac{1}{\cos x} \). This means that if you know the cosine of an angle, you can easily find the secant by taking the reciprocal of that value.
  • For example, if \( \cos x = 0.5 \), then \( \sec x = \frac{1}{0.5} = 2 \).
  • This function is particularly useful when dealing with equations or identities that involve reciprocals of cosine or when transformations require non-zero values (since cosine has zero values where secant is undefined).

A common application of the secant function is in verifying identities and solving trigonometric equations. By substituting \( \sec x = \frac{1}{\cos x} \), equations can often be simplified, as seen in the exercise where \( \sec x - \cos x \) transforms into manageable algebraic operations.
Tangent Function
The tangent function, represented as \( \tan x \), is another vital trigonometric function found by the ratio of the sine and cosine functions. It is expressed as \( \tan x = \frac{\sin x}{\cos x} \). This relationship highlights the tangent as a quotient, making it unique among the primary trigonometric functions.
  • The tangent function is frequently encountered in problems that involve slopes or angles, particularly in calculus and geometry.
  • In our exercise, substituting \( \tan x = \frac{\sin x}{\cos x} \) allows further simplification of the given trigonometric equation.

Additionally, the properties of the tangent function help in transforming and verifying trigonometric identities. By recognizing these properties, such as where the function becomes undefined (points where \( \cos x = 0 \)), students can navigate complex equations with more confidence.
Pythagorean Identity
The Pythagorean identity is one of the core identities in trigonometry, represented by \( \sin^2 x + \cos^2 x = 1 \). This identity is fundamental because it relates the squares of the sine and cosine functions together, reflecting the Pythagorean theorem in a trigonometric context.
  • It provides a tool to simplify expressions and verify identities by replacing one part of the identity with others.
  • For instance, the exercise conveniently uses this identity to show that \( 1 - \cos^2 x = \sin^2 x \), directly aligning with \( \sin^2 x + \cos^2 x = 1 \).

Understanding this identity can significantly ease calculations, particularly in simplifying expressions and solving equations involving squares of trigonometric functions. Recognizing that \( \cos^2 x = 1 - \sin^2 x \), and vice versa, encourages fluency in manipulating and solving trigonometric equations.

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