Problem 55 - Algebra and Trigonometry

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 identity is verified; both sides are equivalent.

Step 1: Transform the Structure of the Left Side

Start by taking the left-hand side of the equation and simplify or restructure. Notice that the form \( \frac{\sin x - 1}{\sin x + 1} \) can potentially be rewritten by altering the numerator. To do this, use the identity \( \sin^2 x + \cos^2 x = 1 \) to express \( \sin x - 1 \) in terms of cosine. Rewrite \( \sin x - 1 = - (1 - \sin x) = -(\cos^2 x)/(\sin x + 1) \), where \( \sin^2 x = 1 - \cos^2 x \).

Step 2: Complete the Denominator Adjustment

Now, let's adjust the denominator of the left-hand side to match the right-hand side. Factor out \((\sin x + 1)\) from both the numerator and denominator: \(\frac{-\cos^2 x}{\sin x + 1} \times \frac{1}{\sin x + 1} \). Thus, the expression becomes \(\frac{-\cos^2 x}{(\sin x + 1)^2}\), which matches the right-hand side of the original identity.

Step 3: Verification

With the transformation and subsequent simplification in Steps 1 and 2, both sides of the identity are now equivalent. The identity \( \frac{\sin x - 1}{\sin x + 1} = \frac{-\cos^2 x}{(\sin x + 1)^2} \) is verified because simplifying the left-hand side without loss of generality matches the right-hand side.

Key Concepts

Cosine Function

The cosine function is one of the fundamental functions in trigonometry, commonly denoted as \( \cos(x) \). It is essential for understanding the relationships within right triangles and periodic phenomena in mathematics and physics.
In a right triangle, the cosine of an angle \( \theta \) is defined as the ratio of the length of the adjacent side to the hypotenuse. Mathematically, this can be written as:

• \( \cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}} \)

The cosine function is periodic with a period of \( 2\pi \), meaning that the function repeats every \( 2\pi \) radians. Additionally, it has important properties such as:

• Even function: \( \cos(-x) = \cos(x) \)
• Range: [-1, 1]

In the context of the given exercise, recognizing the expression \( \cos^2(x) \) is crucial in transforming and simplifying the identity. This squared function emerges directly from the Pythagorean identity, which is pivotal for the given simplification.

Sine Function

The sine function is another core element of trigonometry, written as \( \sin(x) \). Like the cosine function, it helps describe the ratios of sides in right triangles and represents oscillations.
In right triangle terminology, the sine of an angle \( \theta \) is the ratio of the length of the opposite side to the hypotenuse:

• \( \sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}} \)

Just as with the cosine function, the sine function has a period of \( 2\pi \) and follows these properties:

• Odd function: \( \sin(-x) = -\sin(x) \)
• Range: [-1, 1]

In solving trigonometric identities like the one in the problem, the function \( \sin(x) \) is manipulated to facilitate simplification. The exercise illustrates how using this function in combination with the Pythagorean identity can simplify complex expressions.

Pythagorean Identity

The Pythagorean identity is a fundamental equation in trigonometry that links the sine and cosine functions together: \[ \sin^2 x + \cos^2 x = 1 \] This relationship arises from the Pythagorean theorem and is indispensable in simplifying trigonometric expressions. It is particularly useful when you need to express one trigonometric function in terms of another, as is often required in proving identities.
For instance, in the exercise where you need to verify the identity involving \( \frac{\sin x - 1}{\sin x + 1} \), the Pythagorean identity allows you to express \( \sin^2 x \) in terms of \( \cos^2 x \):

• \( \sin^2 x = 1 - \cos^2 x \)

Utilizing this identity enables the transformation and manipulation necessary to simplify and prove the desired trigonometric identity. It essentially helps bridge the gap between different trigonometric functions within an expression.

Trigonometric Simplification

Trigonometric simplification is the process of transforming a complex trigonometric expression into a simpler form. This often involves the use of identities, algebraic manipulation, and logical reasoning.
The goal is typically to make an expression easier to handle, more straightforward to understand, or to prove an identity. Here’s a basic approach to trigonometric simplification:

• Identify trigonometric identities that can simplify the expression.
• Rewrite complex functions using known identities.
• Combine like terms and simplify fractions when possible.

In the exercise provided, trigonometric simplification is used to verify the given identity. Transforming the left-hand side to match the right-hand side through algebraic manipulation and strategic use of the Pythagorean identity showcases the power of simplification in proving equations strong and valid. By re-expressing \( \sin x - 1 \) and considering the denominator, the problem guides you through an insightful process of reducing complexity while maintaining equivalence.