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Magazine Chapter 1: Problem 151
Verify that each equation is an identity. \(\quad \cot \gamma+\tan \gamma=\sec \gamma \csc \gamma\)
Short Answer
Expert verified
The equation is an identity as both sides simplify to \( \frac{1}{\sin \gamma \cos \gamma} \).
Step by step solution
01
Express in terms of sine and cosine
The first step in verifying this identity is to express all trigonometric functions in terms of sine and cosine. Recall that \( \cot \gamma = \frac{\cos \gamma}{\sin \gamma} \), \( \tan \gamma = \frac{\sin \gamma}{\cos \gamma} \), \( \sec \gamma = \frac{1}{\cos \gamma} \), and \( \csc \gamma = \frac{1}{\sin \gamma} \).Thus, the left side of the equation becomes: \( \frac{\cos \gamma}{\sin \gamma} + \frac{\sin \gamma}{\cos \gamma} \) and the right side becomes: \( \frac{1}{\cos \gamma} \cdot \frac{1}{\sin \gamma} \).
02
Simplify the Left Side
Now, let's simplify each part of the equation. Starting with the left side: \( \frac{\cos \gamma}{\sin \gamma} + \frac{\sin \gamma}{\cos \gamma} \). To combine these fractions, we need a common denominator:\[ \frac{\cos^2 \gamma + \sin^2 \gamma}{\sin \gamma \cos \gamma} \]Using the Pythagorean identity: \( \cos^2 \gamma + \sin^2 \gamma = 1 \), the left side simplifies further to:\[ \frac{1}{\sin \gamma \cos \gamma} \]
03
Simplify the Right Side
Now examine the right side: \( \frac{1}{\cos \gamma \sin \gamma} \). This is already in a simplified form and matches exactly what we derived for the left side.
04
Conclude the Identity
Since both sides are equal after simplification, \( \frac{1}{\sin \gamma \cos \gamma} = \frac{1}{\cos \gamma \sin \gamma} \), we have verified that the given equation \( \cot \gamma + \tan \gamma = \sec \gamma \csc \gamma \) is indeed an identity.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Cotangent
The concept of cotangent is foundational in trigonometry, and it can be understood as the reciprocal of tangent. For any angle \( \gamma \), the cotangent function is defined as \( \cot \gamma = \frac{\cos \gamma}{\sin \gamma} \). This means that cotangent is the ratio of the cosine of an angle to its sine.
Here are some points to help you remember:
Here are some points to help you remember:
- Cotangent is undefined when \( \sin \gamma = 0 \), since dividing by zero is not possible.
- It is often used in calculus and geometry to simplify trigonometric expressions.
- Cotangent, like other trigonometric functions, is periodic and will repeat its values over a full cycle of \( 2\pi \) radians.
Tangent
The tangent function is one of the primary trigonometric functions, crucial for solving various mathematical problems. In terms of sine and cosine, it can be represented as \( \tan \gamma = \frac{\sin \gamma}{\cos \gamma} \).
**Key Notes on Tangent:**
**Key Notes on Tangent:**
- Tangent shows how sine and cosine relate through division, making it valuable in right triangle calculations.
- It is undefined wherever \( \cos \gamma = 0 \), as dividing by zero is not allowed.
- Its values become quite large when \( \gamma \) approaches the points where cosine is near zero.
Secant
The secant function, denoted as \( \sec \gamma \), is less commonly discussed than sine and cosine, but is still very useful. It is defined as the reciprocal of cosine: \( \sec \gamma = \frac{1}{\cos \gamma} \).
**Things to Remember about Secant:**
**Things to Remember about Secant:**
- It is undefined when \( \cos \gamma = 0 \), similar to how the tangent function behaves.
- Secant helps in simplifying expressions in mathematical proofs and identities.
- It becomes very large as \( \cos \gamma \) nears zero, which mirrors the behavior of tangent.
Cosecant
Cosecant, symbolized by \( \csc \gamma \), is the reciprocal of the sine function, and thus can be written as \( \csc \gamma = \frac{1}{\sin \gamma} \).
**Fundamental Insights into Cosecant:**
**Fundamental Insights into Cosecant:**
- This function is undefined wherever \( \sin \gamma = 0 \), coinciding with zeros in the sine function.
- It is primarily used in trigonometric identities and integrals, which require reciprocals.
- Cosecant grows very large as sine approaches zero, similar to how secant behaves relative to cosine.
Pythagorean Identity
The Pythagorean identity is one of the most essential properties in trigonometry. It states that for any angle \( \gamma \), the equation \( \cos^2 \gamma + \sin^2 \gamma = 1 \) holds true.
**Why the Pythagorean Identity is Important:**
**Why the Pythagorean Identity is Important:**
- It provides a check-point and simplification tool across various trigonometric problems.
- This identity is equivalent to the Pythagorean theorem in trigonometric terms.
- It enables transformations and manipulations in complex expressions, allowing one to express any two of these functions in terms of the third one.
