Problem 30 - Contemporary Precalculus

Question

State whether or not the equation is an identity. If it is an identity, prove it. $$\sec (-x)=\sec x$$

Step-by-Step Solution

Answer

If so, prove it. Answer: Yes, the equation is an identity. We proved it using the properties of the secant and cosine functions. The key step was to recognize the even nature of the cosine function, which allowed us to show that $$\sec(-x) = \frac{1}{\cos(-x)} = \frac{1}{\cos(x)} = \sec(x)$$.

Step 1: Recall the definition of secant function

The secant function is defined as the reciprocal of the cosine function, that is: $$\sec(x) = \frac{1}{\cos(x)}$$

Step 2: Analyze the behavior of secant function under change of sign

Let's analyze the behavior of secant function when we change the sign of its argument: $$\sec(-x) = \frac{1}{\cos(-x)}$$ Now, recall the property of cosine function, which is an even function, that states: $$\cos(-x) = \cos(x)$$

Step 3: Substitute the property of cosine function

Now, substitute the property of cosine function in the expression of secant function: $$\sec(-x) = \frac{1}{\cos(-x)} = \frac{1}{\cos(x)} = \sec(x)$$ Since we have shown that the given equation is true for all values of x, we can conclude that it is an identity, and we have proven it as well.

Key Concepts

Secant Function

The secant function is a fundamental trigonometric function that plays a crucial role in various mathematical problems. It's defined formally as the reciprocal of the cosine function:
\[ \sec(x) = \frac{1}{\cos(x)} \]
This means that, wherever the cosine function is non-zero, the secant function is simply the flipped version of it. The secant is undefined where the cosine equals zero, such as \( x = \frac{\pi}{2}, \frac{3\pi}{2}, ... \). This is because division by zero is undefined. The secant function, therefore, must be understood largely in relation to its reciprocal nature, offering a unique perspective into trigonometric problems.

Cosine Function

The cosine function is one of the primary trigonometric functions, closely related to the concepts of waves and circles. It's denoted by \( \cos(x) \), and its values oscillate between -1 and 1. One core property is its periodicity; this function repeats every \( 2\pi \) radians.
One particularity of the cosine function, as shown in our original exercise, is that it's an even function. This means:

• \( \cos(-x) = \cos(x) \)
• This symmetry about the \( y \)-axis helps in simplifying many trigonometric identities, as negative angles have the same cosine value as their positive counterparts.

Understanding the symmetric nature of the cosine function is vital, especially when proving identities such as \( \sec(-x) = \sec(x) \).

Even Functions

An even function is a type of function where its graph is mirrored symmetrically about the \( y \)-axis. This means that for any input \( x \), the function satisfies \( f(-x) = f(x) \). The most illustrative example of an even function in trigonometry is the cosine function.
Even functions are incredibly useful because their symmetry simplifies computations and helps in solving equations. When dealing with even functions, you can often replace negative inputs with positive ones without changing the function's value. This property is pivotal when verifying or proving trigonometric identities—as seen in the identity \( \sec(-x) = \sec(x) \). In this context, knowing that \( \cos(-x) = \cos(x) \) backed by its even nature makes the evaluation straightforward.

Reciprocal Function

Reciprocal functions are relationships where one quantity is the inverse of another. In terms of trigonometry, understanding reciprocal functions involves grasping how one function transforms into another when flipped. The secant, cosecant, and cotangent functions are examples of reciprocals of cosine, sine, and tangent functions, respectively.
For instance, the secant function is the reciprocal of the cosine function, given by \( \sec(x) = \frac{1}{\cos(x)} \). This reciprocal relationship implies that as the cosine function approaches zero, the secant function approaches infinity, showing a strong relationship between these two functions. Reciprocal functions are essential in calculus and trigonometry, as they open a window into understanding limits, asymptotes, and behavior of functions in different intervals.