Problem 18 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-50: Verify the identity. $$ \frac{\cot x}{\csc x+1}=\frac{\csc x-1}{\cot x} $$

Step-by-Step Solution

Answer

The identity is verified since both sides are equal after simplification.

Step 1: Simplify the left-hand side (LHS)

The LHS of the equation is \( \frac{\cot x}{\csc x + 1} \). Recall that \( \cot x = \frac{\cos x}{\sin x} \) and \( \csc x = \frac{1}{\sin x} \). Thus, the LHS becomes \( \frac{\frac{\cos x}{\sin x}}{\frac{1}{\sin x} + 1} = \frac{\frac{\cos x}{\sin x}}{\frac{1+\sin x}{\sin x}} \), which simplifies to \( \frac{\cos x}{1+\sin x} \).

Step 2: Simplify the right-hand side (RHS)

The RHS is \( \frac{\csc x - 1}{\cot x} \), which can be rewritten using \( \csc x = \frac{1}{\sin x} \) and \( \cot x = \frac{\cos x}{\sin x} \) as \( \frac{\frac{1}{\sin x} - 1}{\frac{\cos x}{\sin x}} = \frac{\frac{1-\sin x}{\sin x}}{\frac{\cos x}{\sin x}} \). This simplifies to \( \frac{1-\sin x}{\cos x} \).

Step 3: Show equivalence of LHS and RHS

To prove \( \frac{\cos x}{1+\sin x} = \frac{1-\sin x}{\cos x} \), cross-multiply: \( (\cos x)^2 = (1 - \sin x)(1 + \sin x) \). Expand the right side: \( (1 - \sin x)(1 + \sin x) = 1 - (\sin x)^2 \), which is \( (\cos x)^2 \). Hence, both sides are equal, and the identity is verified.

Key Concepts

Understanding Cotangent

The cotangent function, often denoted as \( \cot x \), is a fundamental trigonometric identity that you encounter when studying angles and triangles.
Let's break it down:

• Definition of Cotangent: By definition, \( \cot x \) is the reciprocal of the tangent function. If \( \tan x = \frac{\sin x}{\cos x} \), then \( \cot x = \frac{\cos x}{\sin x} \).
• Relation with the Unit Circle: On the unit circle, cotangent represents the ratio of the x-coordinate to the y-coordinate of a point corresponding to a specific angle.
• When is it Undefined? Cotangent is undefined when \( \sin x = 0 \), because you cannot divide by zero. This occurs at integer multiples of \( \pi \) (e.g., at angles 0, \( \pi \), etc.).

Understanding and manipulating \( \cot x \), as seen in the exercise, helps simplify trigonometric expressions and solve equations by converting cotangent into sines and cosines, bridging more complex trigonometric identities.

Exploring Cosecant

Cosecant, represented as \( \csc x \), is another key trigonometric identity that's essential for simplifying expressions and solving equations.
Here's what you need to know:

• Definition of Cosecant: The cosecant is the reciprocal of the sine function. Therefore, \( \csc x = \frac{1}{\sin x} \).
• Relation with the Circle: While dealing with the unit circle, cosecant gives the length of the hypotenuse divided by the opposite side for a given angle.
• When is it Undefined? Cosecant is undefined when \( \sin x = 0 \). These points align with where cotangent is undefined, at multiples of \( \pi \).

In trigonometric identities, expressing terms with \( \csc x \) often allows you to convert them into more manageable functions like sine and cosine, as demonstrated in the original solution.

Simplifying Trigonometric Expressions

Simplifying expressions involving trigonometric identities can make complex problems more approachable. The main goal of simplification is to transform a given expression into its simplest form using known identities.
Here’s how the simplification process works:

• Convert to Sine and Cosine: Many trigonometric problems become simpler when rewritten in terms of \( \sin x \) and \( \cos x \). For example, using conversions like \( \cot x = \frac{\cos x}{\sin x} \) or \( \csc x = \frac{1}{\sin x} \).
• Look for Patterns: Look for common identities, such as the Pythagorean identities, which are used to simplify expressions further. For instance, \( \sin^2 x + \cos^2 x = 1 \).
• Cross-Multiplication: When faced with fractions, cross-multiply to check if two expressions are equal. This step can help prove identities, as demonstrated when proving the equivalence of the left-hand and right-hand sides of the given expression.

By mastering these techniques, you build a strong foundation for solving trigonometric equations and verifying identities.