Problem 49 - Contemporary Precalculus

Question

State whether or not the equation is an identity. If it is an identity, prove it. $$\frac{1+\sin x}{1-\sin x}=\frac{\sec x+\tan x}{\sec x-\tan x}$$

Step-by-Step Solution

Answer

Answer: Yes, the given equation is an identity.

Step 1: Identify Trigonometric Identities

First, recognize that \(\sec x = \frac{1}{\cos x}\) and \(\tan x = \frac{\sin x}{\cos x}\). We will use these identities to simplify the right-hand side (RHS) of the equation.

Step 2: Simplify the RHS of the Equation

Using the above mentioned identities, rewrite the RHS of the equation as follows: $$\frac{\sec x+\tan x}{\sec x-\tan x} = \frac{\frac{1}{\cos x}+\frac{\sin x}{\cos x}}{\frac{1}{\cos x}-\frac{\sin x}{\cos x}}$$

Step 3:Combine the fractions and Further Simplify the RHS

Now that we have both terms in each numerator and denominator with the same denominator, we can combine their numerators: $$\frac{\frac{1+\sin x}{\cos x}}{\frac{1-\sin x}{\cos x}}$$

Step 4: Cancel the Denominators

We can simplify this fraction by multiplying both the numerator and the denominator of the RHS by \(\cos x\), effectively canceling out the \(\cos x\) terms in both the numerators and denominators. $$\frac{1+\sin x}{1-\sin x}$$ We can see that both sides of the equation are now equal (

Step 5: State the Conclusion

Since the equation has been simplified to the same expression on both the left-hand side (LHS) and right-hand side (RHS), we can conclude that the given equation is indeed an identity: $$\frac{1+\sin x}{1-\sin x}=\frac{\sec x+\tan x}{\sec x-\tan x}$$

Key Concepts

Identity Proof

In trigonometry, an identity is an equation that holds true for all values within its domain. To establish if an equation is an identity, we typically manipulate one side of the equation using known trigonometric identities until it matches the other side. Such proofs bolster our understanding of trigonometric relationships and often simplify solving more complex problems.
For example, in the given exercise, the equation \( \frac{1+\sin x}{1-\sin x}=\frac{\sec x+\tan x}{\sec x-\tan x} \) is proven to be an identity. By simplifying the right-hand side to match the left-hand side, we confirm that both sides of the equation are equivalent for all permissible values of \( x \).
This process not only involves algebraic manipulation but also requires careful selection of known identities like those for secant and tangent. It's a logical journey that enhances comprehension of the symmetry and elegance inherent in trigonometric functions.

Secant Function

The secant function, denoted as \( \sec x \), is the reciprocal of the cosine function. This means it's defined as \( \sec x = \frac{1}{\cos x} \).
Understanding the secant function is crucial in trigonometry because it's less intuitive than sine or cosine, yet it appears frequently in identities and integrals.

• Secant is undefined where \( \cos x = 0 \), such as \( x = \frac{\pi}{2} + k\pi \), where \( k \) is an integer.
• Since secant is the reciprocal of cosine, it inherits its periodic nature with a period of \( 2\pi \).

In the provided exercise, rewriting \( \sec x \) as \( \frac{1}{\cos x} \) allows simplification to demonstrate the identity. The rearrangement helps in visualizing and manipulating the components of the trigonometric equation.

Tangent Function

The tangent function, represented as \( \tan x \), is a fundamental trigonometric function defined as the ratio of sine to cosine: \( \tan x = \frac{\sin x}{\cos x} \). This function has some compelling features which make it vital in various trigonometric identities.

• Unlike sine and cosine, tangent's period is \( \pi \), meaning it repeats itself after every \( \pi \) interval.
• Tangent is undefined where the cosine function is zero, that is at \( x = \frac{\pi}{2} + k\pi \).

In practicing problems like the given exercise, replacing \( \tan x \) with its equivalent \( \frac{\sin x}{\cos x} \) is essential for identity proofs. This step exposes the intertwined nature of trigonometric functions and aids in the simplification process.

Simplification of Expressions

Simplification of expressions is a critical skill in mathematics, particularly in trigonometry where complex expressions involving multiple functions often need to be reduced to a simpler form.
Here are steps to simplify expressions effectively:

• First, identify any known identities, like \( \sec x = \frac{1}{\cos x} \) or \( \tan x = \frac{\sin x}{\cos x} \), which can transform the equation.
• Next, combine terms where possible. Common denominators and like terms, as seen in the simplification of fractions, help in reducing complexity.
• Then, cancel out common factors. In rational expressions, common factors in both numerator and denominator can be canceled, making the expression easier to handle and comprehend. This was illustrated by eliminating the \( \cos x \) terms in our problem.

Simplification invariably enhances understanding and provides a clearer picture of the relationship between different components of the expression. In our identity proof, these steps lead to showing that both sides of the equation were fundamentally the same.