Problem 23 - Precalculus with Limits

Question

In Exercises 9-50, verify the identity \( \dfrac{\cot x}{\sec x} = \csc x - \sin x \)

Step-by-Step Solution

Answer

After breaking down Cotangent, Secant and Cosecant into terms of Sine and Cosine and performing the correct substitution, it is confirmed that \( \dfrac{\cot x}{\sec x} = \csc x - \sin x \), so the given identity holds true.

Step 1: Break down Cotangent and Secant into terms of Sine and Cosine

The given identity is \( \dfrac{\cot x}{\sec x} = \csc x - \sin x \). Let's break down the left side into more fundamental trigonometric functions. So, \( \cot x = \dfrac{\cos x}{\sin x} \) and \( \sec x = \dfrac{1}{\cos x} \). Hence, the left side becomes \( \dfrac{\dfrac{\cos x}{\sin x}}{\dfrac{1}{\cos x}} \).

Step 2: Simplify Left Hand Side

Simplify the above expression by multiplying the numerator and the denominator by \( \cos x \). We get \( \dfrac{\cos^2 x}{\sin x} \). Now we know that \( \cos^2 x = 1 -\sin^2 x \), so we can replace \( \cos^2 x \) with \( 1 -\sin^2 x \). Now the expression becomes \( \dfrac{1 -\sin^2 x}{\sin x} \). We can break this down to: \( \dfrac{1}{\sin x} - \sin x \).

Step 3: Recognize Cosecant and Simplify

The step from here to the right hand side is to recognize that \( \dfrac{1}{\sin x} = \csc x \). So we substitute \( \dfrac{1}{\sin x} \) with \( \csc x \) and simplify the expression. This will give us \( \csc x - \sin x \) which is equal to the right hand side of the given identity. Hence, the given identity \( \dfrac{\cot x}{\sec x} = \csc x - \sin x \) is indeed true.

Key Concepts

Cotangent

Cotangent is one of the six primary trigonometric functions. It is often abbreviated as "cot" and is defined as the reciprocal of the tangent function. The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. Therefore, the cotangent of an angle is given by the formula:

• \(\cot x = \dfrac{1}{\tan x} = \dfrac{\cos x}{\sin x}\)

This means cotangent expresses the ratio of the adjacent side to the opposite side in a right triangle. In the context of the unit circle, where a point corresponding to an angle \( x \) is given by \((\cos x, \sin x)\), cotangent is the ratio of the x-coordinate (cosine) to the y-coordinate (sine).
Recognizing the reciprocal relationships between trigonometric functions is crucial. For example, knowing that \( \cot x = \dfrac{1}{\tan x} \), helps us easily transform and manipulate expressions involving cotangent.

Secant

The secant function, abbreviated as "sec," is another essential trigonometric function. It is the reciprocal of the cosine function. In a right triangle, the cosine of an angle is the ratio of the adjacent side to the hypotenuse. Hence, secant is defined as:

• \(\sec x = \dfrac{1}{\cos x}\)

Secant represents the ratio of the hypotenuse to the adjacent side. In unit circle terms, it denotes how many times larger the hypotenuse is compared to the adjacent side of the angle. Because it is the reciprocal, secant takes values greater than or equal to 1 for angles where cosine is non-zero. Recognizing secant in identities can assist in simplifying expressions and solving equations involving trigonometric functions.
For example, in our exercise, transforming the expression \( \dfrac{\cot x}{\sec x} \) required us to break down our functions into sine and cosine terms. By doing so, it simplifies the process of verifying or solving trigonometric identities.

Cosecant

Cosecant is the last of the three reciprocal trigonometric functions we will discuss. Often abbreviated as "csc," it is the reciprocal of the sine function. In a right triangle, the sine is the ratio of the opposite side to the hypotenuse. Therefore, the cosecant is:

• \(\csc x = \dfrac{1}{\sin x}\)

This means that cosecant gives the ratio of the hypotenuse to the opposite side. On the unit circle, it represents the multiplicative inverse of the y-coordinate, which is sine. As a result, like secant, cosecant values are greater than 1 when sine is not zero, since it involves the hypotenuse which is longer than the opposite side.
Cosecant often appears in identities and expressions alongside other functions, facilitating simplification. In the identity we verified, recognizing that \( \dfrac{1}{\sin x} = \csc x \) was key in confirming that the left and right sides of the equation matched. This highlights the role of reciprocal identities in solving trigonometric problems.