Problem 49 - Algebra and Trigonometry

Question

Verify the identity: $$\cot x+\tan x=\csc x \sec x$$

Step-by-Step Solution

Answer

After simplification and using the Pythagorean identity, both sides of the equation are the same so the identity \(\cot x + \tan x = \csc x \sec x\) is verified.

Step 1: Express everything in terms of sine and cosine

This means: \(\cot x + \tan x = \csc x \sec x \Longrightarrow \frac{\cos x}{\sin x} + \frac{\sin x}{\cos x} = \frac{1}{\sin x} \cdot \frac{1}{\cos x}\).

Step 2: Simplify both sides of the equation

Make the denominators the same on the left side. This implies multiplying the first term by \(\frac{\cos x}{\cos x}\) and the second term by \(\frac{\sin x}{\sin x}\). The left side becomes \(\frac{\cos^2 x}{\sin x \cos x} + \frac{\sin^2 x}{\sin x \cos x}\) and the right side simplifies to \(\frac{1}{\sin x \cos x}\). The equation is now \(\frac{\cos^2 x + \sin^2 x}{\sin x \cos x} = \frac{1}{\sin x \cos x}\).

Step 3: Use the Pythagorean identity

The Pythagorean identity states that \(\sin^2 x + \cos^2 x = 1\), therefore, the left side simplifies to \(\frac{1}{\sin x \cos x}\). Now, both sides of the equation are the same: \(\frac{1}{\sin x \cos x} = \frac{1}{\sin x \cos x}\).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the variables involved, provided that the expressions are well-defined. They are vital tools in the fields of calculus, geometry, and all types of wave motion analysis. For instance, in verifying the identity \(\cot x + \tan x = \csc x \sec x\), we make use of these identities to simplify complex expressions.

One key approach is recognizing that complex trigonometric functions like cotangent, secant, and cosecant can be rewritten in terms of the more fundamental functions sine and cosine. This is useful for both solving trigonometric equations and verifying identities, as it often simplifies the expressions and makes the underlying relationships more apparent.

When dealing with such problems, it's important to remember that the goal is not to prove something new, but rather to show that what's given can be transformed into an equivalent statement using known identities. This is often achieved step by step, manipulating each side of the equation separately until they look the same.

Pythagorean Identity

The Pythagorean identity is one of the most fundamental and widely used trigonometric identities. It states that for any angle \( x \), \( \sin^2 x + \cos^2 x = 1 \). This relationship emerges from the Pythagorean theorem applied to the unit circle, where the length of the hypotenuse is 1.

We see the Pythagorean identity in action when we simplify complex trigonometric expressions. In the earlier exercise of verifying \(\cot x + \tan x = \csc x \sec x\), this identity helps collapse \(\cos^2 x + \sin^2 x\) into 1, which is crucial for equating the two sides of the identity. It's instrumental for the simplification in our problem-solving process.

This identity is versatile and can lead to other key identities when combined or manipulated with algebra—for example, deriving the expressions for \(1 - \sin^2 x\) and \(1 - \cos^2 x\), which are also useful in solving and verifying trigonometric identities.

Cotangent and Tangent

Understanding Cotangent and Tangent

In trigonometry, cotangent (\(\cot\)) and tangent (\(\tan\)) are functions that describe the ratios of sides in a right triangle relative to a specific angle, aside from their definitions on the unit circle.

The cotangent of an angle is the ratio of the adjacent side to the opposite side, which is the reciprocal of the tangent function that represents the ratio of the opposite side to the adjacent side. In terms of sine and cosine, the cotangent is \(\frac{\cos}{\sin}\), and tangent is \(\frac{\sin}{\cos}\).

Using Cotangent and Tangent

In practice, you can use these relationships to transform equations and expressions. During the verification of the identity \(\cot x + \tan x = \csc x \sec x\), these functions are expressed in their sine and cosine form to facilitate a common basis for simplification. Not only does this technique make the algebraic manipulation more straightforward, but it also sets the stage for invoking other identities like the Pythagorean identity.

Sine and Cosine

Sine (\(\sin\)) and cosine (\(\cos\)) are the foundational trigonometric functions from which all other trigonometric functions can be derived. They relate the angles of a triangle to the ratios of two sides and are intrinsic to the unit circle representation, where they correspond to the y and x coordinates, respectively.

For solving trigonometric equations and proving identities, expressing functions in terms of sine and cosine is frequently the first step. Doing so allows us to employ a common language for all trigonometric functions and to apply identities more effectively. For example, in the problem \(\cot x + \tan x = \csc x \sec x\), we began by converting cotangent and tangent to their respective sine and cosine ratios. This set the stage for further simplification and the eventual use of the Pythagorean identity.

Understanding the sine and cosine functions is essential for comprehending the nature of all trigonometric identities. By mastering these basic functions, students can tackle a wide range of trigonometric problems with confidence.