Problem 65 - A Graphical Approach to Precalculus with Limits

Question

Perform indicated operation and simplify the result. $$\frac{1}{\csc ^{2} \theta}+\frac{1}{\sec ^{2} \theta}$$

Step-by-Step Solution

Answer

The simplified result is 1.

Step 1: Rewrite the cosecant and secant

Recall that the cosecant is the reciprocal of the sine, and the secant is the reciprocal of the cosine. So, \[\csc \theta = \frac{1}{\sin \theta} \quad \text{and} \quad \sec \theta = \frac{1}{\cos \theta}.\] Using these definitions, we have:\[ \csc^2 \theta = \frac{1}{\sin^2 \theta} \quad \text{and} \quad \sec^2 \theta = \frac{1}{\cos^2 \theta}. \] Therefore, \[ \frac{1}{\csc^2 \theta} = \sin^2 \theta \quad \text{and} \quad \frac{1}{\sec^2 \theta} = \cos^2 \theta. \]

Step 2: Add the trigonometric expressions

Now that you have the expressions \(\sin^2 \theta\) and \(\cos^2 \theta\), add them together as indicated in the original expression:\[ \sin^2 \theta + \cos^2 \theta. \]

Step 3: Simplify using a trigonometric identity

Recall the Pythagorean identity in trigonometry: \[ \sin^2 \theta + \cos^2 \theta = 1. \] Therefore, the expression simplifies to 1.

Key Concepts

Understanding the Cosecant Function

The cosecant function, noted as \(\csc \theta\), is an important part of trigonometry. It is considered a reciprocal trigonometric function, which means it is the inverse of one of the basic trigonometric functions. Specifically, the cosecant is the reciprocal of the sine function. To understand this better, recall that:

• Sine function (\(\sin \theta\)): This is the ratio of the opposite side to the hypotenuse in a right triangle.
• Cosecant function (\(\csc \theta\)): This is simply the reciprocal of the sine function, defined as \(\csc \theta = \frac{1}{\sin \theta}\).

This means whenever the sine value is known, you can determine the cosecant by taking the reciprocal of that value. In terms of angles, the cosecant is undefined for angles where the sine is zero, because division by zero is undefined.
This occurs, for instance, at angles like 0° and 180°, where the value of \(\sin \theta\) is zero.

Exploring the Secant Function

Much like the cosecant, the secant function is another reciprocal trigonometric function. The secant function, represented as \(\sec \theta\), is the reciprocal of the cosine function.

• Cosine function (\(\cos \theta\)): This function is calculated as the ratio of the adjacent side to the hypotenuse in a right triangle.
• Secant function (\(\sec \theta\)): As the reciprocal function, it is expressed as \(\sec \theta = \frac{1}{\cos \theta}\).

Understanding the secant function is about recognizing that it 'flips' the cosine value. Consequently, wherever the cosine is zero, the secant will be undefined.
This happens at angles like 90° and 270°, because \(\cos \theta\) becomes zero and you cannot divide by zero. The secant function, like cosecant, is a pillar in understanding more complex trigonometric identities and equations.

Diving into the Pythagorean Identity

Among the cornerstone identities in trigonometry is the Pythagorean identity. This identity helps simplify expressions that involve square trigonometric terms.
The identity is written as:\[ \sin^2 \theta + \cos^2 \theta = 1 \]This formula is derived from the Pythagorean theorem applied to a unit circle. The unit circle has a radius of one, and each trigonometric function relates to a side of a right triangle. Here’s how it helps:

• Use it to simplify sums of squared sine and cosine expressions.
• A fundamental identity that provides relationships among trigonometric functions.

Whenever you encounter complex expressions that involve trigonometric terms squared, check whether this identity can simplify it. For example, in the exercise, adding \(\sin^2 \theta\) and \(\cos^2 \theta\) allows us to directly use this identity to simplify to one.

Unpacking Reciprocal Trigonometric Functions

Reciprocal trigonometric functions are alternatives to the primary trigonometric functions and offer useful mathematical relationships.

• The sine, cosine, and tangent functions have their reciprocals as the cosecant, secant, and cotangent, respectively.

Reciprocal functions are useful because they allow us to explore relationships and solve equations involving angles and length ratios where direct calculations would be cumbersome.
Whenever you see trigonometric functions, keep the reciprocal relationships in mind:

• \(\csc \theta = \frac{1}{\sin \theta}\)
• \(\sec \theta = \frac{1}{\cos \theta}\)
• \(\cot \theta = \frac{1}{\tan \theta}\)

Remember that these functions will be undefined if their base functions are zero. Keeping these relationships in mind simplifies working with trigonometric expressions and equations. They open the door to solving many trigonometric problems that seem complex at first glance.