Problem 57 - Precalculus Mathematics for Calculus

Question

Verify the identity. $$\frac{\sin x-1}{\sin x+1}=\frac{-\cos ^{2} x}{(\sin x+1)^{2}}$$

Step-by-Step Solution

Answer

The given identity is verified; both sides simplify to \( \frac{\sin x - 1}{\sin x + 1} \).

Step 1: Simplify the Left Side Expression

Let's start with simplifying the left-hand side of the equation \( \frac{\sin x-1}{\sin x+1} \). This expression cannot be simplified any further, so let's hold it for now.

Step 2: Simplify the Right Side Expression

Now let us simplify the right-hand side of the equation \( \frac{-\cos^2 x}{(\sin x+1)^2} \). The expression contains \( -\cos^2 x \), which can be rewritten using the Pythagorean identity \( \cos^2 x = 1 - \sin^2 x \). Thus, \( -\cos^2 x = \sin^2 x - 1 \). Substituting this into the equation yields:\[ \frac{\sin^2 x - 1}{(\sin x + 1)^2} \]

Step 3: Compare Simplified Expressions

Now that we have \( \frac{\sin x-1}{\sin x+1} \) on the left and \( \frac{\sin^2 x - 1}{(\sin x + 1)^2} \) on the right, let's notice that the numerator \( \sin^2 x - 1 \) can also be written as \((\sin x - 1)(\sin x + 1)\) using the identity \( a^2 - b^2 = (a-b)(a+b) \). Substitute this back:\[ \frac{(\sin x - 1)(\sin x + 1)}{(\sin x + 1)^2} \]

Step 4: Finalize and Verify the Identity

The expression in Step 3, \( \frac{(\sin x - 1)(\sin x + 1)}{(\sin x + 1)^2} \), simplifies to \( \frac{\sin x - 1}{\sin x + 1} \) since \( \sin x + 1 \) in the numerator and \( \sin x + 1 \) in the denominator cancel each other out, where \( \sin x + 1 eq 0 \). Thus we have reached the same expression on both sides, verifying that the original identity holds.

Key Concepts

Pythagorean Identity

The Pythagorean Identity is one of the fundamental building blocks in trigonometry. It relates the square of the sine and cosine functions to the number one. This identity is expressed as:\[ \sin^2 x + \cos^2 x = 1 \]This equation stems from the relationship between the side lengths of a right triangle and the circle's radius in a unit circle framework. In the original exercise, the Pythagorean Identity was crucial for transforming the right side of the equation. By recognizing that \( \cos^2 x \) can be rewritten as \( 1 - \sin^2 x \), we can flip the expression for \( -\cos^2 x \) to \( \sin^2 x - 1 \). This allowed the expression to be restructured in a form that aided in further simplifications. Understanding this identity, thus, empowers one to see beyond the apparent form of trigonometric expressions and to manipulate them skillfully.

Simplifying Expressions

Simplifying expressions requires the application of algebraic identities and manipulations to reduce more complex forms into simpler ones. It involves rewriting expressions so that certain parts can cancel out or combine more effectively, making the math clearer and more elegant.In our example, although the left-hand side \( \frac{\sin x - 1}{\sin x + 1} \) was already in its simplest form, the right side required simplification.

• Transforming \( -\cos^2 x \) using the Pythagorean Identity to become \( \sin^2 x - 1 \).
• Recognizing that \( \sin^2 x - 1 \) is a difference of squares, which factors into \((\sin x - 1)(\sin x + 1)\).

These steps helped to match both sides of the trigonometric identity, showing how simplification can lead to equivalent expressions.

Verifying Identities

Verifying an identity involves proving that two sides of an equation are equivalent for all values within the domain. It goes beyond showing the equality for specific values, promoting a universal truth in mathematics. To verify, follow these steps:

• Simplify each side of the given identity separately. Sometimes one side simplifies further than the other, revealing an underlying relationship or pattern.
• Look for opportunities to employ known identities like the Pythagorean Identity, sum and difference formulas, or factorization techniques.
• Manipulate and compare both expressions, ensuring every step logically follows from the previous one.

In the exercise above, we simplified each side until both appeared in the same form, thus verifying the identity. This process helps build understanding of the interplay of trigonometric functions and deepens comprehension of mathematical equivalences.