Problem 28 - Algebra and Trigonometry

Question

Verify the identity. $$ \frac{\cot x \sec x}{\csc x}=1 $$

Step-by-Step Solution

Answer

The identity is verified, as the expression simplifies to 1.

Step 1: Understand the Trigonometric Identities

To verify the given identity, we need to use trigonometric identities. Recall that \( \cot x = \frac{1}{\tan x}, \sec x = \frac{1}{\cos x}, \) and \( \csc x = \frac{1}{\sin x} \). These identities will help us rewrite the given expression in simpler terms.

Step 2: Substitute Trigonometric Identities

Substitute \( \cot x = \frac{\cos x}{\sin x} \), \( \sec x = \frac{1}{\cos x} \), and \( \csc x = \frac{1}{\sin x} \) into the left side of the given equation: \[ \frac{\cot x \sec x}{\csc x} = \frac{\left( \frac{\cos x}{\sin x} \right) \left( \frac{1}{\cos x} \right)}{\frac{1}{\sin x}} \].

Step 3: Simplify the Expression

Simplify the expression from Step 2. The \( \cos x \) in the numerator and the denominator cancels out, which gives:\[ \frac{\frac{1}{\sin x}}{\frac{1}{\sin x}} = \frac{1}{1} \].

Step 4: Simplified Result

The expression simplifies to \( 1 \). This matches the right side of the original identity, confirming that the identity \( \frac{\cot x \sec x}{\csc x}=1 \) is valid.

Key Concepts

Cotangent

Understanding cotangent can be simple if we break it down into parts. Cotangent, abbreviated as "cot," is one of the six fundamental trigonometric functions. It's the reciprocal of the tangent function. This means that while tangent is defined as the ratio of the opposite side to the adjacent side for a given angle in a right triangle, cotangent is defined as the adjacent side over the opposite side.
The formula for cotangent is given by:

• Cotangent in terms of tangent: \( \cot x = \frac{1}{\tan x} \)
• Cotangent in terms of sine and cosine: \( \cot x = \frac{\cos x}{\sin x} \)

This dual definition means that cotangent can be used in different ways to simplify trigonometric expressions, especially in identities. It's important to know that cotangent is undefined where the sine value is zero because it involves dividing by sine.

Secant

Secant, abbreviated as "sec," is another trigonometric function that's pivotal in simplifying expressions. Secant is the reciprocal of the cosine function. In simple terms, it flips the cosine function; instead of adjacent over hypotenuse (as cosine is), secant is hypotenuse over adjacent.
Its formula is expressed as:

• Secant: \( \sec x = \frac{1}{\cos x} \)

The secant function is undefined where the cosine value becomes zero because it would involve division by zero. Just like other trigonometric functions, secant has its range and properties that are useful when solving trigonometric equations or verifying identities. Remembering its relationship with cosine helps in switching between the forms, making simplification easier.

Cosecant

The function called cosecant, abbreviated as "csc," is closely related to secant. It is the reciprocal of the sine function. This means instead of opposite over hypotenuse (the definition of sine), cosecant is hypotenuse over opposite.
It can be formulated as:

• Cosecant: \( \csc x = \frac{1}{\sin x} \)

Cosecant is undefined where the sine equals zero, which is an important point to consider. Cosecant often appears in trigonometric identities to help transform expressions, and understanding how to switch between cosecant and sine is critical. The friendly relationship between these functions offers a pathway for simplification and understanding of more complex problems.

Trigonometric Simplification

Trigonometric simplification involves translating complex trigonometric expressions into simpler forms using identities. This usually means employing the reciprocal identities, quotient identities, and Pythagorean identities.
For the example in the solution:

• First, use the reciprocal relationships, like \( \cot x = \frac{\cos x}{\sin x} \), \( \sec x = \frac{1}{\cos x} \), and \( \csc x = \frac{1}{\sin x} \)
• Simplify the combination of these expressions by canceling common factors
• This leads to a more manageable expression

The goal of simplification is to transform the equation until it matches another expression or reveals a clearer form of itself. By focusing on one part at a time and using known identities, complex questions like verifying the identity \( \frac{\cot x \sec x}{\csc x}=1 \) become much more approachable.