Question

Use Fundamental Identities and/or the Complementary Angle Theorem to find the exact value of each expression. Do not use a calculator. $$\cot 25^{\circ}-\frac{\cos 25^{\circ}}{\sin 25^{\circ}}$$

Short answer

The final value is 0.

Step-by-step solution

Identify the Trigonometric Functions

Understand the given expression: \ \( \cot 25^{\circ} - \frac{\cos 25^{\circ}}{\sin 25^{\circ}} \ \). Identify that \ \( \cot 25^{\circ} = \frac{1}{\tan 25^{\circ}} \ \) and \ \( \frac{\cos 25^{\circ}}{\sin 25^{\circ}} = \cot 25^{\circ} \ \).

Rewrite the Expression

Rewrite the given expression using the identified identities: \ \( \cot 25^{\circ} - \cot 25^{\circ} \ \). This simplifies to: \ \( 0 \ \).

Key concepts

Cotangent

The cotangent function, often abbreviated as 'cot,' is one of the basic trigonometric functions. It is defined as the reciprocal of the tangent function. If you know a bit about tangent (tan), which is the ratio of the opposite side to the adjacent side in a right triangle, then cotangent will seem familiar.

Mathematically, it's expressed as:
\[ \text{cot}(\theta) = \frac{1}{\text{tan}(\theta)} = \frac{\text{adjacent}}{\text{opposite}} \]

For instance, if you have an angle of 25 degrees, the cotangent of 25 degrees is the reciprocal of the tangent of 25 degrees:
\[ \text{cot}(25^\text{circ}) = \frac{1}{\text{tan}(25^\text{circ})} \]

Understanding cotangent is essential in our exercise because it shows up in the given expression. By recognizing that \( \frac{\text{cos}(25^\text{circ})}{\text{sin}(25^\text{circ})} \) is also equal to \( \text{cot}(25^\text{circ}) \), it makes the process of simplification clearer.

Complementary Angles

Complementary angles are two angles whose sum is 90 degrees. This concept is quite important in trigonometry because trigonometric functions of complementary angles are related in specific ways.

The most common relationships are:

• \( \text{sin}(90^\text{circ} - \theta) = \text{cos}(\theta) \)
• \( \text{cos}(90^\text{circ} - \theta) = \text{sin}(\theta) \)
• \( \text{tan}(90^\text{circ} - \theta) = \text{cot}(\theta) \)

In the exercise, although we didn't directly use the complementary angle theorem, understanding these relationships can be beneficial. For example, knowing that \( \text{tan}(65^\text{circ}) = \text{cot}(25^\text{circ}) \) helps us appreciate the structure and inherent relationships within trigonometric functions.

Trigonometric Simplification

Trigonometric simplification involves using identities and known values to reduce complex expressions to simpler forms.

Let's break down the given expression:
\[ \text{cot}(25^\text{circ}) - \frac{\text{cos}(25^\text{circ})}{\text{sin}(25^\text{circ})} \]

Step by Step Simplification:
First, recall the identity for cotangent:
\[ \text{cot}(25^\text{circ}) = \frac{1}{\text{tan}(25^\text{circ})} = \frac{\text{cos}(25^\text{circ})}{\text{sin}(25^\text{circ})} \]

With this information, rewrite the expression:
\[ \text{cot}(25^\text{circ}) - \text{cot}(25^\text{circ}) \]

This instantly simplifies to:
\[ 0 \]

The key to mastering trigonometric simplification is to know your identities and recognize them quickly. By doing so, you can save yourself time and avoid common pitfalls in problems like these.