Problem 57 - Precalculus with Limits

Question

In Exercises 37 - 58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer. \( \cot u \sin u + \tan u \cos u \)

Step-by-Step Solution

Answer

The simplified form of the expression \( \cot u \sin u + \tan u \cos u \) is \( \cos u + \sin u \).

Step 1: Identify the fundamental identities

The fundamental identities for cotangent and tangent are: \[ \cot u = \frac {\cos u}{\sin u} \] and \[ \tan u = \frac {\sin u}{\cos u} \] respectively.

Step 2: Substitute the identities into the expression

By substituting the identities for cotangent and tangent into the expression, it becomes \[ \frac {\cos u}{\sin u} \cdot \sin u + \frac {\sin u}{\cos u} \cdot \cos u \]

Step 3: Simplify the expression

On simplification, the expression becomes \[ \cos u + \sin u \]. As there is no simplified form for this expression, we'll leave it in this state.

Key Concepts

Understanding Cotangent

The cotangent function is one of the core trigonometric functions. It is the reciprocal of the tangent function. Knowing the relationship between cotangent and tangent helps us simplify trigonometric expressions.
When we discuss cotangent, we are referring to the ratio of the adjacent side to the opposite side of a right-angled triangle. It is given by:

• \( \cot u = \frac{\cos u}{\sin u} \)

In this expression, \( u \) is the angle, \( \cos u \) is the cosine of the angle (adjacent over hypotenuse), and \( \sin u \) is the sine of the angle (opposite over hypotenuse).
Using the relationship of cotangent, we can transform and manipulate expressions, as seen in the given exercise. By replacing \( \cot u \) with its identity, transforming or simplifying expressions becomes straightforward. This is key when trying to reduce more complex trigonometric expressions.

The Role of Tangent in Trigonometry

Tangent is another fundamental trigonometric function, providing us with the ratio of the opposite side to the adjacent side of a triangle.
The identity associated with the tangent function can simplify many expressions, including the one from our exercise. It is given by:

• \( \tan u = \frac{\sin u}{\cos u} \)

By substituting \( \tan u \) in an expression, complex components often become manageable.
This reciprocal relationship between angle sides is useful in converting trigonometric expressions into simpler forms, which is a common need in mathematics problems.
For example, when transforming \( \tan u \) in an expression by its identity, the calculations focus more directly on the inherent trigonometric ratios, ultimately making the final expression easier to interpret.

Simplifying Trigonometric Expressions

Simplifying expressions is essential in mathematics to obtain a clearer or more practical form. For trigonometric expressions, this involves using identities to reduce complexity.
In our example, the expression \( \cot u \sin u + \tan u \cos u \) was simplified by substituting the fundamental identities of cotangent and tangent.
After substitution, the expression becomes\[ \frac{\cos u}{\sin u} \cdot \sin u + \frac{\sin u}{\cos u} \cdot \cos u \].

• The \( \frac{\cos u}{\sin u} \cdot \sin u \) simplifies to \( \cos u \).
• The \( \frac{\sin u}{\cos u} \cdot \cos u \) simplifies to \( \sin u \).

Combining these results, we end up with the simplified form of \( \cos u + \sin u \).
Throughout this process, clarity and simplification help us understand the relationships between different components of trigonometric expressions, aiding in both solving and understanding mathematical problems.