Problem 37 - Precalculus: A Concise Course

Question

Use trigonometric identities to transform the left side of the equation into the right side \((0<\theta<\pi / 2)\). $$ \tan \theta \cot \theta=1 $$

Step-by-Step Solution

Answer

The transformation results in 1.

Step 1: Identifying Trigonometric Identities

Recall the following trigonometric identities: \( \tan\theta = \frac{\sin\theta}{\cos\theta} \) and \( \cot\theta = \frac{\cos\theta}{\sin\theta} \).

Step 2: Substituting in Equation

Substitute these identities into the given equation: \( \tan\theta \cot\theta = \frac{\sin\theta}{\cos\theta} \cdot \frac{\cos\theta}{\sin\theta} \).

Step 3: Simplifying the Equation

After substituting, the \(\sin\theta\) in the numerator cancels out the \(\sin\theta\) in the denominator. Similarly, the \(\cos \theta\) in the denominator cancels out the \(\cos\theta\) in the numerator. The final result is 1.

Key Concepts

Understanding the Tangent

The tangent function, denoted as \( \tan \theta \), is a fundamental concept in trigonometry. It arises from the ratio of the two sides of a right triangle that are opposite and adjacent to an angle \( \theta \). This gives us the formula:

• \( \tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} \)

In trigonometric terms, this can be expressed using the sine and cosine functions, as \( \tan\theta = \frac{\sin\theta}{\cos\theta} \). This identity is crucial because it allows us to express the tangent in terms of more basic trigonometric functions.

When working with tangent, it is important to note that it is undefined for angles where \( \cos\theta = 0 \), such as \( 90^\circ, 270^\circ, \text{etc.} \), because division by zero is undefined. Understanding these properties helps in manipulating and simplifying trigonometric expressions, like those in the given exercise.

Exploring the Cotangent

The cotangent function, \( \cot \theta \), is another important trigonometric ratio. It is the reciprocal of the tangent function and can be understood as the inverse of \( \tan \theta \). The formula for cotangent is:

• \( \cot \theta = \frac{1}{\tan \theta} \)
• This can also be expressed as \( \cot\theta = \frac{\cos\theta}{\sin\theta} \)

The cotangent, just like tangent, is undefined where its denominator becomes zero, in this case, where \( \sin\theta = 0 \), which occurs at \( 0^\circ, 180^\circ, \text{etc.} \).

Understanding cotangent is vital in transforming expressions like \( \tan\theta \cdot \cot\theta \) into simpler forms. Observing how tangent and cotangent interact allows for elegant simplifications, as demonstrated by their product equating to 1 in the exercise.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions involves using identities to rewrite complex expressions in a simpler or more useful form. In the realm of trigonometry, identities are equations that hold true for all valid values of the involved variables.

For the given problem, we demonstrated simplification by using the identities:

• \( \tan\theta = \frac{\sin\theta}{\cos\theta} \)
• \( \cot\theta = \frac{\cos\theta}{\sin\theta} \)

By substituting these into the equation \( \tan\theta \cdot \cot\theta \), the result is \( \frac{\sin\theta}{\cos\theta} \times \frac{\cos\theta}{\sin\theta} \), which simplifies to 1 due to the cancellation of \( \sin\theta \) and \( \cos\theta \) from the numerator and denominator.

This example showcases the power of trigonometric identities in simplifying expressions, reinforcing a deeper understanding of their interactions and aiding in solving more complex trigonometric equations.