Problem 12 - Precalculus with Limits

Question

In Exercises 9-50, verify the identity \( \cos x + \sin x \tan x = \sec x \)

Step-by-Step Solution

Answer

The identity \( \cos x + \sin x \tan x = \sec x \) is indeed correct provided \( \cos x ≠ 0 \).

Step 1: Rewrite in terms of sin and cos

To start off, all terms of trigonometric functions in the given identity should be written in terms of sine (sin) and cosine (cos). The given identity is: \( \cos x + \sin x \tan x = \sec x \). Since, \( \tan x \) is equal to \( \frac{\sin x}{\cos x} \) and \( \sec x \) is equal to \( \frac{1}{\cos x} \), replace these terms in the identity to obtain: \( \cos x + \sin x \frac{\sin x}{\cos x} = \frac{1}{\cos x} \).

Step 2: Simplify equation

Next, the equation is simplified. This starts with multiplying the second term on the left side of the equation by \(\frac{\cos x}{\cos x}\). This gives: \( \cos x + \sin^2 x = \frac{1}{\cos x} \). Now, using the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\), replace the \(\sin^2 x\) in the previous equation with \(1 - \cos^2 x\). So, it now transforms to: \( \cos x + 1 - \cos^2 x = \frac{1}{\cos x} \). Further, rearrange the equation to: \( 1 = \cos^2 x + \cos x + \frac{1}{\cos x}\).

Step 3: Factorize

To simplify it further, multiply the entire equation by \( \cos x \). This gives: \( \cos x = \cos^3 x + \cos^2 x + 1 \). This equation can be simplified to \( \cos^3 x + \cos^2 x - \cos x + 1 = 0 \). This equation can be factorized to give: \( \cos x (\cos^2 x + \cos x - 1) + 1 = 0 \). The factor \( \cos x \) can be factored out to give \( \cos x (\cos x + 1)^2 = 0 \). This equation is true since the square of any real number is always positive, it's evident that the expression is only zero when \( \cos x = 0 \)!

Step 4: Validate the identity

To validate the given identity (without letting \( \cos x = 0 \)), reduce the equation \( \cos x (\cos x + 1)^2 = 0 \) to \( \cos x = 0 \). This brings back the working equation to the initial given identity \( \cos x + \sin x \tan x = \sec x \). Therefore, for all \( x \) such that \( \cos x ≠ 0 \), the given identity is validated.

Key Concepts

Sine and Cosine

Sine (\( \sin \)) and cosine (\( \cos \)) are the fundamental building blocks of trigonometry. These functions relate the angles of a right triangle to the ratios of its sides. Whenever we deal with any trigonometric identity, it is often beneficial to express all terms in terms of sine and cosine to see their interrelations clearly.

- **Sine** of an angle is the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle. - **Cosine** represents the ratio of the length of the adjacent side to the hypotenuse.

These identities are crucial in simplifying trigonometric expressions. In the exercise, we saw how expressing tangent as a fraction (\( \tan x = \frac{\sin x}{\cos x} \)) and secant (\( \sec x = \frac{1}{\cos x} \)) in terms of cosine helped break down and validate the identity. Understanding these basic relationships simplifies more complex equations.

Tangent and Secant

The tangent (\( \tan \)) and secant (\( \sec \)) functions are built from the foundational sine and cosine functions. They extend the relationships within a right triangle to form additional trigonometric identities.

- **Tangent** (\( \tan x \)) is the ratio of the sine to the cosine of an angle. Mathematically, it's expressed as \( \tan x = \frac{\sin x}{\cos x} \). This makes tangent a crucial function for converting expressions in terms of sine and cosine.

- **Secant** (\( \sec x \)) is simply the reciprocal of the cosine function. It is given by \( \sec x = \frac{1}{\cos x} \). Secant is often used when you need to turn division by cosine into multiplication.
Both tangent and secant are indispensable in transforming and proving trigonometric identities. The given exercise illustrates this by using these functions to reframe the identity and facilitate verification.

Pythagorean Identities

The Pythagorean identities are powerful tools utilized in solving and verifying trigonometric equations. These identities stem from the Pythagorean theorem, connecting the sides of right triangles in trigonometric functions.

There are three key Pythagorean identities:

• \( \sin^2 x + \cos^2 x = 1 \)
• \( 1 + \tan^2 x = \sec^2 x \)
• \( 1 + \cot^2 x = \csc^2 x \)

The first identity, \( \sin^2 x + \cos^2 x = 1 \), is particularly useful in our exercise. It enabled substituting \( \sin^2 x \) with \( 1 - \cos^2 x \) to further simplify the identity, illustrating its value in transformation processes.

Utilizing these identities not only simplifies problems but also allows for additional relationships and proofs to be discovered within trigonometric contexts. Understanding and applying them is fundamental to mastering trigonometry.