Problem 105 - A Graphical Approach to Precalculus with Limits

Question

Verify that equation is an identity. \(\frac{1-\sin \theta}{1+\sin \theta}=\sec ^{2} \theta-2 \sec \theta \tan \theta+\tan ^{2} \theta\)

Step-by-Step Solution

Answer

Yes, the given equation is an identity.

Step 1: Simplify the right side

The right side is \(\sec^2\theta - 2\sec\theta\tan\theta + \tan^2\theta\). Substituting \(\sec\theta = \frac{1}{\cos\theta}\) and \(\tan\theta = \frac{\sin\theta}{\cos\theta}\):
\(= \frac{1}{\cos^2\theta} - \frac{2\sin\theta}{\cos^2\theta} + \frac{\sin^2\theta}{\cos^2\theta} = \frac{1 - 2\sin\theta + \sin^2\theta}{\cos^2\theta}\)

Step 2: Factor the numerator

The numerator \(1 - 2\sin\theta + \sin^2\theta = (1 - \sin\theta)^2\).
So the right side becomes \(\frac{(1-\sin\theta)^2}{\cos^2\theta}\).

Step 3: Use the Pythagorean identity

Since \(\cos^2\theta = 1 - \sin^2\theta = (1-\sin\theta)(1+\sin\theta)\):
\(\frac{(1-\sin\theta)^2}{(1-\sin\theta)(1+\sin\theta)} = \frac{1-\sin\theta}{1+\sin\theta}\)

Step 4: Conclusion

The right side simplifies to \(\frac{1-\sin\theta}{1+\sin\theta}\), which equals the left side. Therefore, the equation is an identity.

Key Concepts

Secant

The secant function, denoted as \( \sec \theta \), is a fundamental trigonometric function closely related to the cosine function. It is defined as the reciprocal of the cosine function:- \( \sec \theta = \frac{1}{\cos \theta} \)This means that whenever you need to find the value of the secant, you take the value of cosine and divide 1 by it. However, remember that the cosine value must not be zero, as division by zero is undefined.
The secant function is particularly useful in various applications, especially in solving trigonometric equations or verifying identities. Since it involves the reciprocal relationship with cosine, changes in \( \cos \theta \) correspond to changes in \( \sec \theta \).
For example, if \( \cos \theta = \frac{1}{2} \), then \( \sec \theta = 2 \). This reciprocal relationship allows for unique transformations and simplifications in trigonometric expressions, making it crucial in exercises like the one presented here.

Tangent

Tangent, denoted as \( \tan \theta \), is another key trigonometric function. It is defined by the ratio of sine to cosine:- \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)In unit-circle terms, tangent represents the slope of the line at any angle \( \theta \), indicating how steeply it moves away from the horizontal. This slope is particularly impactful when working with trigonometric identities, as it allows intricate angle manipulations.

In our exercise, the tangent function transforms into squares and cross terms like \( \tan^2 \theta \) and \( 2\sec\theta\tan\theta \), emphasizing its flexible nature. Understanding tangent's relationship with the other primary trigonometric functions unveils many identity verifications.
Therefore, knowing how to convert between these forms equips you to tackle more complex equations or substitution tasks that feature both tangent and secant, as seen in the given problem.

Trigonometric Equations

Solving trigonometric equations often involves replacing functions with their identities to simplify expressions and find solutions. A trigonometric equation consists of trigonometric functions like sine, cosine, tangent, and secant. These might appear complex, but familiarity with identities eases this task significantly.

In our original exercise, you need to show that both sides of the equation are equal using known identities:

• Secant: \( \sec \theta = \frac{1}{\cos \theta} \)
• Tangent: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

The solution provided steps through simplifying the right-hand expression by substituting these identities. Like fractions, trigonometric expressions can often be combined into a single term. This allows the comparison with other fractions or simpler expressions.
Solving such equations can be a systematic process. Replacing everything by their base-trigonometric relationships is crucial. And just like in regular algebra, you look to simplify until you reach a form comparable to the other side of the equation. This problem specifically deals with verifying an identity, thus illustrating how a combination of algebra and trigonometry can validate such mathematical truths.