Problem 50 - Algebra 2

Question

Verify that each of the following is an identity. \(\frac{\sec \theta}{\tan \theta}=\csc \theta\)

Step-by-Step Solution

Answer

The identity is verified as both sides simplify to \( \csc \theta \).

Step 1: Recall Definitions

First, recall the definitions of the trigonometric functions involved. The secant function is defined as \( \sec \theta = \frac{1}{\cos \theta} \), the tangent function is \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), and the cosecant function is \( \csc \theta = \frac{1}{\sin \theta} \).

Step 2: Substitute and Simplify the Left Side

Substitute the definitions of \( \sec \theta \) and \( \tan \theta \) into the left-hand side of the equation: \( \frac{\sec \theta}{\tan \theta} = \frac{\frac{1}{\cos \theta}}{\frac{\sin \theta}{\cos \theta}} \). Simplify this expression by multiplying by the reciprocal: \( \frac{1}{\cos \theta} \times \frac{\cos \theta}{\sin \theta} = \frac{1}{\sin \theta} \).

Step 3: Compare with Right Side

Observe that the simplified left-hand side \( \frac{1}{\sin \theta} \) is identical to the right-hand side \( \csc \theta \). Since both sides are equal, the given equation is verified as an identity.

Key Concepts

Secant Function

The secant function, denoted as \( \sec \theta \), is a fundamental trigonometric function that is intimately related to the cosine function. The secant function is essentially the reciprocal of the cosine function.
It is defined as:

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

Understanding this relationship is crucial. In simple terms, wherever we have a cosine function, the secant is the flipped version in terms of reciprocal value. This characteristic is helpful in various trigonometric identities and equations.
For instance, in any right triangle with an angle \( \theta \), if you imagine a unit circle, the secant function represents the length of a line segment from the center of the circle to a point on the terminal side of the angle, projected along the radius or "hypotenuse equivalent". It is a way to think of the secant geometrically.
Be careful, though, as \( \sec \theta \) is undefined wherever \( \cos \theta = 0 \), meaning \( \theta \) is an odd multiple of \( \frac{\pi}{2} \). So, \( \theta = \frac{\pi}{2}, \frac{3\pi}{2}, ... \) would make the secant function undefined. This is important to remember when solving problems.

Tangent Function

The tangent function, denoted as \( \tan \theta \), is another key trigonometric function that ties together the sine and cosine functions. It tells us the ratio of the sine to the cosine of an angle in a right triangle.
The tangent is defined as:

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

You can visualize \( \tan \theta \) by thinking of it as the slope of the line formed by the angle \( \theta \) in the unit circle. In the context of a right triangle, \( \tan \theta \) represents the ratio of the opposite side to the adjacent side when considering angle \( \theta \).
Like the secant function, \( \tan \theta \) has points where it is undefined. Whenever \( \cos \theta \) is zero, \( \tan \theta \) becomes undefined because division by zero isn't possible. Hence, \( \tan \theta \) is undefined at angles where \( \theta = \frac{\pi}{2}, \frac{3\pi}{2}, ... \), similar to the secant function.
This function plays an essential role in various applications, such as determining the angle of elevation or depression in real-world scenarios.

Cosecant Function

The cosecant function, denoted as \( \csc \theta \), is the reciprocal counterpart of the sine function. Where the sine function represents the ratio of the opposite side over the hypotenuse, the cosecant flips this relationship.
Its definition is straightforward:

• \( \csc \theta = \frac{1}{\sin \theta} \)

This definition indicates that wherever \( \sin \theta \) is a small value, \( \csc \theta \) becomes relatively larger, emphasizing the reciprocal nature. Like its fellow reciprocal functions, \( \csc \theta \) is undefined when \( \sin \theta \) is zero, which occurs at angles where \( \theta = 0, \pi, 2\pi, ... \).
Geometrically, if you imagine the unit circle, the cosecant function can help deduce the vertical projection length from the hypotenuse to the opposite side. It's an abstract concept but very useful when breaking down complex identities.
In proving trigonometric identities, as shown in the exercise, expressing \( \frac{\sec \theta}{\tan \theta} \) as \( \csc \theta \) can simplify the equation, unifying both sides of an identity. Mastering these relationships enhances problem-solving efficiency.