Problem 74 - Algebra and Trigonometry

Question

Verify the identity. $$ \frac{\cos \theta}{1-\sin \theta}=\sec \theta+\tan \theta $$

Step-by-Step Solution

Answer

Identity verified: both sides simplify to \( \frac{1 + \sin \theta}{\cos \theta} \).

Step 1: Express Secant and Tangent in Terms of Sine and Cosine

First, express both \( \sec \theta \) and \( \tan \theta \) in terms of sine and cosine. Recall that \( \sec \theta = \frac{1}{\cos \theta} \) and \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). So, the right side becomes \( \frac{1}{\cos \theta} + \frac{\sin \theta}{\cos \theta} = \frac{1 + \sin \theta}{\cos \theta} \).

Step 2: Obtain a Common Denominator on the Right Side

Since both terms on the right side of the equation have \( \cos \theta \) in the denominator, combine them into a single fraction: \( \frac{1 + \sin \theta}{\cos \theta} \).

Step 3: Simplify the Left Side by Multiplying Numerator and Denominator

To simplify \( \frac{\cos \theta}{1 - \sin \theta} \), multiply the numerator and the denominator by \( 1 + \sin \theta \). This results in \( \frac{\cos \theta (1 + \sin \theta)}{(1 - \sin \theta)(1 + \sin \theta)} \).

Step 4: Apply the Difference of Squares Formula in the Denominator

The denominator \( (1 - \sin \theta)(1 + \sin \theta) \) simplifies using the difference of squares to \( 1 - \sin^2 \theta \), which is equal to \( \cos^2 \theta \) due to the Pythagorean identity. Thus, \( \frac{(1 - \sin^2 \theta)}{\cos^2 \theta} = \frac{\cos^2 \theta}{\cos^2 \theta} \).

Step 5: Simplify the Resulting Expression

Now simplify the expression to \( \frac{\cos \theta (1 + \sin \theta)}{\cos^2 \theta} = \frac{1 + \sin \theta}{\cos \theta} \) after cancelling \( \cos \theta \) in the numerator and denominator.

Step 6: Conclude the Verification

Both sides of the equation simplify to \( \frac{1 + \sin \theta}{\cos \theta} \), verifying the identity. Therefore, \( \frac{\cos \theta}{1 - \sin \theta} = \sec \theta + \tan \theta \).

Key Concepts

Secant Function

The secant function, denoted as \( \sec \theta \), is a vital trigonometric function. It's defined as the reciprocal of the cosine function. This means

• \( \sec \theta = \frac{1}{\cos \theta} \)

The secant function is essential in trigonometry for solving equations and verifying identities like the one in the problem provided. By understanding that the secant is simply another way to express terms involving cosine, we can transform complex expressions to help reveal underlying structures or simplify them. In the exercise, this understanding allows us to transform and manipulate the equation efficiently while holding the meaning intact.

Tangent Function

The tangent function, represented as \( \tan \theta \), describes the ratio of sine to cosine:

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

This function plays a crucial role in many trigonometric identities, including the one we are tackling. Because it converts sine and cosine into a single ratio, it often simplifies calculations, especially when combined with the secant. By translating \( \tan \theta \) into sine and cosine, problems can transition from abstract expressions to more manageable algebraic form, as demonstrated when combining \( \sec \theta + \tan \theta \) into a single fraction in the step-by-step solution.

Pythagorean Identity

The Pythagorean identity is one of the most foundational identities in trigonometry, stating that

• \( \sin^2 \theta + \cos^2 \theta = 1 \)

This powerful equation allows us to express one trigonometric function in terms of another, simplifying the evaluation within equations and identities. In the provided solution, the Pythagorean identity facilitates simplification by transforming \( 1 - \sin^2 \theta \) into \( \cos^2 \theta \). This step of simplifying the denominator using the identity not only shows the relevance of this concept but also highlights its frequent use in verifying and solving trigonometric expressions. It is a key tool for resolving differences of squares, as applied in this exercise.

Difference of Squares

The difference of squares formula is a well-known algebraic identity that states:

• \( a^2 - b^2 = (a-b)(a+b) \)

This concept is particularly useful for simplifying products of binomials, such as in the expression \((1 - \sin \theta)(1 + \sin \theta)\). Applying this identity yields \( 1 - \sin^2 \theta \), which is crucial for simplifying trigonometric equations, as seen in this solution. Once re-expressed using the Pythagorean identity, it eloquently transposes into \( \cos^2 \theta \), confirming that mathematics involves interconnected logical steps that unravel the complexities of algebra and trigonometry alike. Utilizing these principles means transforming multifaceted problems into solvable equations that follow familiar trigonometric rules.