Problem 45 - Precalculus: A Concise Course

Question

Use trigonometric identities to transform the left side of the equation into the right side \((0<\theta<\pi / 2)\). $$ \frac{\sin \theta}{\cos \theta}+\frac{\cos \theta}{\sin \theta}=\csc \theta \sec \theta $$

Step-by-Step Solution

Answer

The original equation \(\frac{\sin θ}{\cos θ} + \frac{\cos θ}{\sin θ} = \csc θ \sec θ\) is indeed equal as after using multiple trigonometric identities on the left side and the right side we received \(\sec^2 θ = \sec^2 θ\). So, the equation holds true.

Step 1: Apply Trigonometric Identities

We start by rewriting the left hand side (LHS) in terms of cotangent and tangent. Using the identities \(\tan θ = \frac{\sin θ}{\cos θ}\) and \(\cot θ = \frac{\cos θ}{\sin θ}\), the equation can be rewritten as \(\tan θ + \cot θ\).

Step 2: Using Reciprocal Trigonometric Identities

Knowing that \(\cot θ = \frac{1}{\tan θ}\), the LHS becomes \(\tan θ + \frac{1}{\tan θ}\). Now let's multiply the terms by \(\tan θ\) for simplification. After multiplying we have \(\tan^2 θ + 1\).

Step 3: Applying Pythagorean Identity

We use the Pythagorean identity, \(\tan^2 θ + 1 = \sec^2 θ\), so we have \(\sec^2 θ\) on the LHS.

Step 4: Write the Right-Hand Side in terms of secant

Now, we rewrite the right-hand side (RHS) in terms of secant and cosecant. We know, \(\csc θ = \frac{1}{\sin θ}\) and \(\sec θ = \frac{1}{\cos θ}\). Stripping these down to basic sine and cosine and knowing that \(\sin θ = \frac{1}{\csc θ}\) and \(\cos θ = \frac{1}{\sec θ}\), the RHS becomes \(\frac{1}{\sin θ\cos θ} = \frac{1}{\tan θ}\). Again applying the identity \(\sec^2 θ = \tan^2 θ + 1\), the equation now becomes \(\sec^2 θ = \sec^2 θ\).

Key Concepts

Pythagorean Identity

Understanding the Pythagorean identity is fundamental in trigonometry. This identity is derived from the Pythagorean theorem, which connects the sides of a right triangle. The Pythagorean identity states that for any angle \(\theta\), the square of the sine function plus the square of the cosine function equals one:
\[\sin^2\theta + \cos^2\theta = 1\]
This identity is pivotal because it shows the inherent relationship between the sine and cosine functions. In practice, it allows us to simplify complex trigonometric expressions. For instance, when confronted with an equation that includes \(\tan^2\theta + 1\), we can apply the Pythagorean identity to transform it into \(\sec^2\theta\), simplifying the equation significantly. This concept is instrumental in solving trigonometric equations, and recognising this identity can make many problems more manageable.

Reciprocal Trigonometric Identities

Reciprocal identities are another key concept in trigonometry, providing a different perspective on the six basic trigonometric functions. These identities define the cosecant \(\csc\), secant \(\sec\), and cotangent \(\cot\) functions as the reciprocals of the sine \(\sin\), cosine \(\cos\), and tangent \(\tan\) functions, respectively. Formally, they are expressed as:
\[\csc \theta = \frac{1}{\sin \theta}\], \[\sec \theta = \frac{1}{\cos \theta}\], and \[\cot \theta = \frac{1}{\tan \theta}\]
These identities simplify the process of manipulating trigonometric expressions. For example, the given exercise uses the reciprocal identities to convert a sum of ratios into the addition of a function and its reciprocal, \(\tan \theta + \frac{1}{\tan \theta}\). Using these identities not only simplifies expressions but can also reveal the fundamental nature of trigonometric relationships.

Tangent and Cotangent

The tangent and cotangent functions have a unique connection in trigonometry. Tangent \(\tan\) is the ratio of the sine to the cosine, and cotangent \(\cot\) is its reciprocal, representing the ratio of the cosine to the sine. These relationships can be expressed as:
\[\tan \theta = \frac{\sin \theta}{\cos \theta}\], \[\cot \theta = \frac{\cos \theta}{\sin \theta}\]
Knowing how to move between tangent and cotangent is beneficial when solving equations like the one from our exercise. In the exercise, the initial equation involved a combination of sine and cosine ratios. Recognizing the tangent and cotangent functions within these ratios was essential in simplifying the equation into something more manageable — essentially transitioning from a sum of ratios to a familiar trigonometric identity that can be simplified further. These two functions often appear in trigonometric identities, simplifications, integrations, and can be particularly useful in modeling cyclic phenomena, such as waves.