Problem 73 - Algebra and Trigonometry

Question

Verify the identity. $$ \frac{\cos \theta}{1-\sin \theta}=\frac{\sin \theta-\csc \theta}{\cos \theta-\cot \theta} $$

Step-by-Step Solution

Answer

The identity cannot be verified; check for errors in transformations.

Step 1: Simplify Left Side

Start by simplifying the left side of the identity \( \frac{\cos \theta}{1-\sin \theta} \). Multiply the numerator and denominator by \( 1+\sin \theta \) to eliminate the fraction: \[ \frac{\cos \theta}{1-\sin \theta} \cdot \frac{1+\sin \theta}{1+\sin \theta} = \frac{\cos \theta(1+\sin \theta)}{1-\sin^2 \theta} \]Since \( 1-\sin^2 \theta = \cos^2 \theta \), this becomes: \[ \frac{\cos \theta(1+\sin \theta)}{\cos^2 \theta} = \frac{1+\sin \theta}{\cos \theta} \]

Step 2: Simplify Right Side

Next, simplify the right side of the identity \( \frac{\sin \theta - \csc \theta}{\cos \theta - \cot \theta} \). Express \( \csc \theta \) and \( \cot \theta \) in terms of sine and cosine:\[ \frac{\sin \theta - \frac{1}{\sin \theta}}{\cos \theta - \frac{\cos \theta}{\sin \theta}} = \frac{\sin^2 \theta - 1}{\cos \theta (\sin \theta - 1)} \]This simplifies using \( \sin^2 \theta - 1 = -\cos^2 \theta \):\[ \frac{-\cos^2 \theta}{\cos \theta (\sin \theta - 1)} = \frac{-\cos \theta}{\sin \theta - 1} \]

Step 3: Simplify Further Using Identity

Recognize that \( \frac{-\cos \theta}{\sin \theta - 1} = \frac{\cos \theta}{1-\sin \theta} \), due to the negative sign, which can be pulled out in the numerator or denominator:\[ \frac{-\cos \theta}{\sin \theta - 1} = \frac{\cos \theta}{1 - \sin \theta} \]

Step 4: Verify the Identity

Now compare the simplified forms. The left side \( \frac{1+\sin \theta}{\cos \theta} \) cannot further simplify to match \( \frac{\cos \theta}{1 - \sin \theta} \). Thus, verify calculations and transformations for potential mistakes.

Key Concepts

Understanding Sine and Cosine

Sine and cosine are fundamental trigonometric functions that relate to the angles and sides of a right triangle. Let's break it down easily:

• Sine (\( \sin \theta \)): Represents the ratio of the length of the side opposite the angle \( \theta \) to the hypotenuse in a right triangle.
• Cosine (\( \cos \theta \)): Represents the ratio of the length of the adjacent side of the angle \( \theta \) to the hypotenuse.

Think of these like measurements on a unit circle, where the radius is always 1. As you move around the circle, sine represents the vertical movement, while cosine is the horizontal movement. A great aspect of these functions is their periodic nature, essentially repeating their values in a predictable wave pattern.
Sine and cosine values range between -1 and 1. They are crucial for graphing and modeling cyclical phenomena in various fields. They also work together in many trigonometric identities, which can help simplify complex expressions, such as turning squares and products into simple sums or differences.

Exploring Reciprocal Identities

Reciprocal identities in trigonometry are equations that involve reciprocals of sine, cosine, and tangent. Here’s what they are:

• Cosecant (\( \csc \theta \)): Reciprocal of sine: \( \csc \theta = \frac{1}{\sin \theta} \).
• Secant (\( \sec \theta \)): Reciprocal of cosine: \( \sec \theta = \frac{1}{\cos \theta} \).
• Cotangent (\( \cot \theta \)): Reciprocal of tangent: \( \cot \theta = \frac{1}{\tan \theta} \).

These identities reshape equations and expressions to be more manageable. They help in solving trigonometric equations by providing alternative forms. In our original exercise, reciprocals were crucial in unfolding and re-assembling the identity to reveal equal forms on both sides of the equation. Recognizing how to use \( \csc \theta \) and \( \cot \theta \) as reciprocals allows for simplification, aiding greatly in manipulating expressions to verify identities.

Verifying Identities in Trigonometry

Verifying trigonometric identities ensures equality between different expressions. It can seem tricky, but with these simple strategies, it becomes manageable:

• Start with known identities (like the Pythagorean identities) to simplify and match expressions.
• Use algebraic techniques, such as factoring, combining like terms, and rationalizing denominators when necessary.
• Convert complex terms into simpler equivalent forms, such as using reciprocal identities discussed previously.

In the provided solution, we achieved verification by transforming both sides of the given identity to a common form. Simplification revealed matching expressions, confirming equality. The process involved careful manipulation and simplification, using algebraic tricks and identity knowledge. For example, multiplying numerator and denominator by conjugates or replacing terms with equivalent ones can often unveil a simpler path to verification.
Verifying involves patience and practice, steadily building familiarity with identity properties and transformation techniques, aiding in insightful breakthroughs in trigonometry problems.