Problem 151 - Calculus Volume 1

Question

Verify that each equation is an identity. \(\quad \cot \gamma+\tan \gamma=\sec \gamma \csc \gamma\)

Step-by-Step Solution

Answer

The equation is an identity as both sides simplify to \( \frac{1}{\sin \gamma \cos \gamma} \).

Step 1: 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: \( \frac{1}{\cos \gamma} \cdot \frac{1}{\sin \gamma} \).

Step 2: 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:\[ \frac{1}{\sin \gamma \cos \gamma} \]

Step 3: 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.

Step 4: 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.

Key Concepts

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:

• 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.

Understanding cotangent shifts perspective from thinking purely about angles, to the relationships between the angles' sine and cosine.

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:**

• 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.

The tangent function is especially important in problems involving angles, slopes, and anywhere rotation or direction needs to be characterized.

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:**

• 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.

Using secant in expressions often requires understanding its behavior relative to cosine, especially how it can simplify expressions when combined with other trigonometric functions.

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:**

• 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.

Understanding cosecant is particularly important when working with identities and equations that involve reducing or solving trigonometric expressions.

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:**

• 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.

Mastery of this identity allows for deeper insights into trigonometry, aiding solutions in both theoretical and applied mathematics contexts.