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Magazine Chapter 20: Problem 18
In Exercises \(7-12,\) multiply and simplify. In Exercises \(13-18,\) factor and simplify. $$\sin x+\sin x \cot ^{2} x$$
Short Answer
Expert verified
The simplified form is \( \csc x \).
Step by step solution
01
Understand the Expression
The initial expression given is \( \sin x + \sin x \cot^2 x \). We need to simplify this by multiplying and then simplifying the result.
02
Apply Trigonometric Identity
Recall the trigonometric identity \( \cot x = \frac{1}{\tan x} \) or \( \cot x = \frac{\cos x}{\sin x} \). Thus, \( \cot^2 x = \left( \frac{\cos x}{\sin x} \right)^2 = \frac{\cos^2 x}{\sin^2 x} \). Substitute this back into the expression.
03
Substitute and Simplify
Replace \( \cot^2 x \) in the expression with \( \frac{\cos^2 x}{\sin^2 x} \). This gives us:\[ \sin x + \sin x \cdot \frac{\cos^2 x}{\sin^2 x} \]Factor out \( \sin x \):\[ \sin x \left( 1 + \frac{\cos^2 x}{\sin^2 x} \right) \].
04
Simplify the Inner Expression
The inner expression is \( 1 + \frac{\cos^2 x}{\sin^2 x} \). Find a common denominator for the terms inside the parentheses:\[ \frac{\sin^2 x}{\sin^2 x} + \frac{\cos^2 x}{\sin^2 x} = \frac{\sin^2 x + \cos^2 x}{\sin^2 x} \].
05
Use Pythagorean Identity
Use the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \). Substitute into the expression:\[ \frac{1}{\sin^2 x} \].
06
Simplify Further
Now substitute this simplified result back into our factored expression:\[ \sin x \cdot \frac{1}{\sin^2 x} \]which simplifies to\[ \frac{1}{\sin x} = \csc x \].
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Simplifying Expressions
When faced with a trigonometric expression like \( \sin x + \sin x \cot^2 x \), the goal is to simplify it step-by-step to make it easier to work with. Begin by examining each part of the expression to identify opportunities to use known identities or algebraic manipulation.
- Identify common terms or factors: In our expression, \( \sin x \) is a common factor.
- Apply identities: Use trigonometric identities, like \( \cot^2 x = \frac{\cos^2 x}{\sin^2 x} \), to express complex parts in simpler terms.
- Combine terms: By factoring out common factors, the expression becomes easier to handle. For example, \( \sin x (1 + \frac{\cos^2 x}{\sin^2 x}) \).
- Simplify fractions and solve: Simplify the expression inside the parentheses by finding a common denominator, making it \( \frac{\sin^2 x + \cos^2 x}{\sin^2 x} \).
Pythagorean Identity
The Pythagorean identity is one of the fundamental tools in trigonometry. It's a simple yet powerful equation: \( \sin^2 x + \cos^2 x = 1 \). In the context of simplifying expressions, this identity often plays a crucial role.
- The Pythagorean identity allows for the replacement of complex trigonometric expressions with simpler ones.
- For the expression \( \sin^2 x + \cos^2 x \), you can replace it with \( 1 \) wherever it appears. This transformation significantly simplifies calculations.
- It aids in reducing formulas: For instance, \( \frac{\sin^2 x + \cos^2 x}{\sin^2 x} \) simplifies to \( \frac{1}{\sin^2 x} \), saving time and minimizing potential errors in computation.
Trigonometric Functions
Trigonometric functions such as sine (\( \sin x \)), cosine (\( \cos x \)), and cotangent (\( \cot x \)) are the building blocks for expressing and manipulating mathematical relationships involving angles.
- Sine and cosine functions define an angle's projection onto the unit circle, relating linear and angular measurements.
- Functions like cotangent, defined as \( \cot x = \frac{\cos x}{\sin x} \), are derivatives of sine and cosine, and they simplify expressions by transforming variables and functions into calculable forms.
- In the exercise, recognizing \( \cot^2 x \) as \( \frac{\cos^2 x}{\sin^2 x} \) allows substitutions that simplify the expression into something more recognizable and manageable.
