Problem 40 - Algebra and Trigonometry

Question

Find all solutions of the equation. $$\sec x-\tan x=\cos x$$

Step-by-Step Solution

Answer

The solutions are \( x = n\pi \) and \( x = \frac{\pi}{2} + 2k\pi \).

Step 1: Use Trigonometric Identities

Start by rewriting the left side of the equation using the identities for secant and tangent. The equation is given as:\[ \sec x - \tan x = \cos x \]Recall that:\[ \sec x = \frac{1}{\cos x} \]\[ \tan x = \frac{\sin x}{\cos x} \]Substituting these identities into the equation gives:\[ \frac{1}{\cos x} - \frac{\sin x}{\cos x} = \cos x \]

Step 2: Combine Expressions on the Left

Combine the terms on the left side of the equation by finding a common denominator:\[ \frac{1 - \sin x}{\cos x} = \cos x \]This simplifies the equation to a single fraction on the left side.

Step 3: Clear the Fraction

Multiply both sides by \( \cos x \) to eliminate the fraction:\[ 1 - \sin x = \cos^2 x \]

Step 4: Use Pythagorean Identity

From the Pythagorean identity, \( \cos^2 x = 1 - \sin^2 x \), substitute this identity into the equation:\[ 1 - \sin x = 1 - \sin^2 x \]

Step 5: Rearrange the Equation

Subtract \(1 - \sin^2 x\) from both sides:\[ - \sin x = - \sin^2 x \]Rewriting gives:\[ \sin^2 x - \sin x = 0 \]

Step 6: Factor the Equation

Factor the quadratic in terms of \( \sin x \):\[ \sin x (\sin x - 1) = 0 \]This gives two solutions: \( \sin x = 0 \) or \( \sin x = 1 \).

Step 7: Solve for x

Solve for \( x \) based on the solutions for \( \sin x \):- \( \sin x = 0 \): The solutions are \( x = n\pi \), where \( n \) is an integer.- \( \sin x = 1 \):The solution is \( x = \frac{\pi}{2} + 2k\pi \), where \( k \) is an integer.

Key Concepts

Secant Function

The secant function, denoted as \( \sec x \), is one of the reciprocal trigonometric functions. It is the reciprocal of the cosine function. Mathematically, it is expressed as:

• \( \sec x = \frac{1}{\cos x} \)

This function is particularly useful when you encounter expressions involving cosine in the denominator because it simplifies calculations.
Secant has a unique pattern, where it becomes undefined at values where cosine is zero (like odd multiples of \( \frac{\pi}{2} \)). These points correspond to the vertical asymptotes of the secant graph.
Whenever you see \( \sec x \), you can think of it as flipping the cosine function, taking its reciprocal to aid in solving complex trigonometric equations. You will often use it alongside other trigonometric identities, as seen in the original step-by-step solution.

Tangent Function

The tangent function, represented as \( \tan x \), emerges from the relationship between sine and cosine. Defined as:

• \( \tan x = \frac{\sin x}{\cos x} \)

This function is widely used in trigonometry to relate angles to ratios in a right triangle. It is also a periodic function, repeating every \( \pi \) radians.
In our exercise, we substitute \( \tan x \) to help simplify the equation. Since the tangent function is undefined where \( \cos x \) is zero, it corresponds to the same points as the secant's vertical asymptotes. These zeroes tie back to how the secant and tangent are interrelated.
Understanding \( \tan x \) is essential for working across various trigonometric problems involving slopes, angles, and periodic patterns.

Pythagorean Identity

The Pythagorean Identity is a cornerstone in trigonometry that relates the squares of the sine and cosine functions:

• \( \sin^2 x + \cos^2 x = 1 \)

You can derive other forms, such as:

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

This identity is vital for transforming and simplifying expressions that involve quadratic trigonometric terms.
In the context of our exercise, the identity allows us to replace \( \cos^2 x \) with \( 1 - \sin^2 x \) to solve the equation further. It's a powerful tool for converting between the sine and cosine functions, helping unravel problems into more manageable forms.
Utilizing the Pythagorean Identity effectively can simplify proofs, calculations, and lead to discovering important trigonometric solutions.