Problem 45 - Precalculus with Limits

Question

In Exercises 37 - 58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer. \( \dfrac{1 - \sin^2 x}{\csc^2 x - 1} \)

Step-by-Step Solution

Answer

The simplified expression is \( \sin^2x \).

Step 1: Substitute the Pythagorean identity

We start off with the Pythagorean identity \( \sin^2x + \cos^2x = 1 \). The expression needs the term \( 1 - sin^2x \), which according to the identity should simply be \( \cos^2x \). So the expression now becomes \( \frac{\cos^2 x}{csc^2 x - 1} \).

Step 2: Convert the Reciprocal identity

In the denominator, we have the term \( \csc^2x \), which is the reciprocal of \( \sin^2x \). Thus, \( \csc^2x = \frac{1}{\sin^2x} \). Replacing this in the denominator and simplifying, the expression becomes \( \frac{\cos^2 x}{\frac{1}{\sin^2 x} - 1} \). This simplifies further to \( \frac{\cos^2 x \sin^2 x}{1 - \sin^2 x} \).

Step 3: Substitute the Pythagorean identity again

Now again use the Pythagorean identity, this time for the denominator of the fraction, replacing \( 1 - \sin^2x \) with \( \cos^2x \). Now, the expression becomes \( \frac{\cos^2 x \sin^2 x}{\cos^2 x} \).

Step 4: Cancel out terms and simplify

The \( \cos^2 x \) term in the numerator and denominator cancel each other out, simplifying the expression in the process. The final, simplified form of the given expression is \( \sin^2 x \).

Key Concepts

Pythagorean Identity

Understanding the Pythagorean identity is vital for simplifying trigonometric expressions. It states that for any angle x, the square of the sine of x and the square of the cosine of x will always add up to 1: \[ \sin^2x + \cos^2x = 1 \.\] This comes directly from the Pythagorean theorem in a right triangle but applies to the unit circle as well. In simplifying trigonometric expressions, this identity is useful because it allows us to replace one trigonometric function with another, potentially simplifying the entire expression.

For instance, in our original problem, we needed to simplify the term \( 1 - \sin^2 x \). Knowing our Pythagorean identity, it was clear that \( \cos^2 x \) was its equivalent. This step is crucial because it transforms the expression into a form where other identities can be applied to continue the simplification process.

Reciprocal Identity

Reciprocal identities are just as important when working with trigonometric expressions as they explain the relationship between certain pairs of trigonometric functions. They are:

• \( \sin x \) and \( \csc x \), where \( \csc x = \frac{1}{\sin x} \)
• \( \cos x \) and \( \sec x \), where \( \sec x = \frac{1}{\cos x} \)
• \( \tan x \) and \( \cot x \), where \( \cot x = \frac{1}{\tan x} \)

Using these identities, we can rewrite trigonometric expressions using their respective reciprocal functions. In the exercise we considered, we applied the reciprocal identity by recognizing that \( \csc^2x \) is the reciprocal of \( \sin^2x \), converting \( \csc^2x \) into \( \frac{1}{\sin^2x} \). This step is a game-changer for simplification because it often reveals opportunities to cancel out terms or combine fractions in a more manageable way.

Trigonometric Simplification

The process of trigonometric simplification involves applying several trigonometric identities to break down complex expressions into their simplest form. The steps typically include:

• Identifying which identities are applicable, such as the Pythagorean identity or reciprocal identity
• Replacing certain trigonometric functions with others to simplify the expression
• Simplifying fractions by combining terms or canceling out common factors

Each step must be taken carefully to prevent mistakes and to ensure the expression is actually being simplified and not just altered in appearance. In our textbook example, after applying the Pythagorean and reciprocal identities, we could transform the given expression into a simple fraction. By paying close attention, we identified that the numerator and denominator had a common factor which could be cancelled out, leaving us with the much simpler \( \sin^2 x \).

Fundamental Identities in Trigonometry

Trigonometry is rich with fundamental identities that offer a toolkit for manipulating and transforming expressions. Besides the Pythagorean and reciprocal identities already discussed, others include angle sum and difference identities, double angle identities, and half-angle identities. Now, while our example problem did not require us to use these additional identities, they are essential in more complex situations.

The understanding of these fundamental identities is critical; they are the building blocks for solving trigonometric equations, proving other theorems, and they even have applications in calculus. It is the interplay among these identities that allows for the clever simplifications seen in tricky trigonometric expressions. Just as a carpenter needs a well-organized toolbox, a student of trigonometry must have a well-prepared set of fundamental identities to draw upon when simplifying expressions and solving problems.