Problem 13 - Algebra and Trigonometry Real Mathematics, Real People

Question

Verify the identity. $$\frac{\csc ^{2} x}{\cot x}=\csc x \sec x$$

Step-by-Step Solution

Answer

The identity \( \frac{\csc ^{2} x}{\cot x} = \csc x \sec x \) is verified through step by step transformations of trigonometric functions to their equivalent form and simplifications.

Step 1: Rewrite cotangent

Start by rewriting the cotangent with its equivalent, which is the reciprocal of tangent. So, \( \cot x \) equals \( \frac{1}{\tan x} \). Therefore, the equation is now \( \frac{\csc ^{2} x}{\frac{1}{\tan x}}= \csc x \sec x \)

Step 2: Simplify the equation

Simplify the equation by removing the fraction in the denominator to obtain \( \csc ^{2} x \cdot \tan x = \csc x \sec x \).

Step 3: Rewrite tangent and secant

Now, change the tangent and the secant expressing them in terms of sine and cosine. The equation becomes \( \csc^{2} x \cdot \frac{\sin x}{\cos x} = \csc x \cdot \frac{1}{\cos x} \).

Step 4: Further simplification

Simplify the left side by cancelling \( \sin x \) and \( \csc x \), and the right side by cancelling \( \cos x \) and \( \csc x \), to obtain \(\csc x = \csc x\).

Step 5: Final Step

Looking at the last step, it's clear that both sides of the equation are equal, therefore, the given identity is verified.

Key Concepts

Cotangent

The cotangent, commonly denoted as \( \cot x \), is one of the six fundamental trigonometric functions. It's defined as the reciprocal of the tangent function. This means that:

• \( \cot x = \frac{1}{\tan x} \)

Taking a closer look, the tangent function itself is the ratio of sine to cosine:

• \( \tan x = \frac{\sin x}{\cos x} \)
• Thus, \( \cot x = \frac{\cos x}{\sin x} \)

Cotangent is often useful in various mathematical simplifications and identities. Knowing its relationship to sine and cosine allows us to manipulate complex expressions more easily. Remembering its definition helps make sense of trigonometric identities and equations.

Cosecant

Cosecant, represented as \( \csc x \), is another reciprocal function. It is the reciprocal of the sine function:

• \( \csc x = \frac{1}{\sin x} \)

This means that when sine is small, cosecant is large, and vice versa. Cosecant is crucial when working with triangles, especially right triangles, since it helps in finding unknown sides. In trigonometric identities, substituting with cosecant can simplify expressions, as seen in the identity we're examining. Using reciprocal relationships streamlines calculations and allows verification of equations more smoothly.

Secant

The secant function, denoted \( \sec x \), is the reciprocal of the cosine function:

• \( \sec x = \frac{1}{\cos x} \)

Secant and cosine are intimately linked, just like sine and cosecant. When cosine is small, the secant value is large. Secant is particularly useful in trigonometric identities. In our example, the expression \( \csc x \sec x \) combines secant with cosecant to simplify verification. Recognizing these reciprocal connections helps to unravel complex trigonometric relationships and makes understanding the equations much easier.

Tangent

Tangent, symbolized as \( \tan x \), relates to both sine and cosine. It is defined by:

• \( \tan x = \frac{\sin x}{\cos x} \)

Tangent provides a direct comparison of the sine and cosine of an angle. It's helpful in understanding angles and slopes, as well as evaluating trigonometric identities. By rewriting tangent in terms of sine and cosine, expressions can be more easily manipulated, aiding in simplification processes. This helps in developing a deeper understanding of trigonometric equations and identities.

Simplification of Fractions

Simplifying fractions is a common task in algebra and trigonometry to make expressions more manageable. This simplification often involves reciprocal identities, multiplication, and cancellation of common terms. In trigonometric problems:

• Combine fractions by finding a common denominator or multiplying by reciprocal values.
• Simplify expressions by cancelling common factors across numerators and denominators.
• Use trigonometric identities to express terms in convenient forms, allowing reduction of complexity.

By following these processes, equations become more straightforward, allowing clearer insights into solving and verifying trigonometric identities like the one given. This simplifies analysis and ensures understanding of the mathematical principles involved.