Problem 35 - Precalculus Mathematics for Calculus

Question

Prove the identity. $$\cot (x-y)=\frac{\cot x \cot y+1}{\cot y-\cot x}$$

Step-by-Step Solution

Answer

The identity is proven by rewriting \( \cot(x-y) \) using trigonometric identities and simplifying to match the given expression.

Step 1: Recall Trigonometric Identities

Recall the trigonometric identities for cotangent. Specifically, that \( \cot(x) = \frac{\cos(x)}{\sin(x)} \) and the identity for \( \cot(x-y) \) which is \( \frac{\cos(x-y)}{\sin(x-y)} \).

Step 2: Rewrite Cotangent of a Difference

The expression for \( \cot(x-y) \) using sine and cosine difference identities is \( \frac{\cos(x-y)}{\sin(x-y)} \). Using the difference identities, we have \( \cos(x-y) = \cos x \cos y + \sin x \sin y \) and \( \sin(x-y) = \sin x \cos y - \cos x \sin y \).

Step 3: Substitute and Simplify

Substitute the identities into the formula for \( \cot(x-y) \):\[ \cot(x-y) = \frac{\cos x \cos y + \sin x \sin y}{\sin x \cos y - \cos x \sin y} \]Now, subdivide each term by \( \sin x \sin y \):\[ \cot(x-y) = \frac{\frac{\cos x \cos y}{\sin x \sin y} + \frac{\sin x \sin y}{\sin x \sin y}}{\frac{\sin x \cos y}{\sin x \sin y} - \frac{\cos x \sin y}{\sin x \sin y}} \]

Step 4: Introduce Cotangent

Recognize that \( \frac{\cos x}{\sin x} = \cot x \) and \( \frac{\cos y}{\sin y} = \cot y \). Substitute these terms:\[ \cot(x-y) = \frac{\cot x \cot y + 1}{\frac{\cos y}{\sin y} - \frac{\cos x}{\sin x}} \] which simplifies to:\[ \cot(x-y) = \frac{\cot x \cot y + 1}{\cot y - \cot x} \]

Step 5: Verification

Check for consistency of transformations. Rather than going through empirical checks, ensure each algebraic manipulation aligns with trigonometric identities. Verify that every step follows logically from the previous step.

Key Concepts

Cotangent Identities

Understanding cotangent identities is crucial for solving trigonometric problems. Cotangent, represented as \( \cot \), is the reciprocal of the tangent function in trigonometry. It is defined as \( \cot(x) = \frac{\cos(x)}{\sin(x)} \).
This identity shows how cotangent is related to sine and cosine, making it easier to manipulate in equations. For example, if you know the cosine and sine of an angle, you can easily find the cotangent.
Here's a simple breakdown:

• \( \cot(x) \) becomes important in solving trigonometric identities, especially when you need to express the relationship in terms of sine and cosine for simplification.
• Knowing \( \cot(x) \) helps in solving identities which involve sums or differences of angles due to its connection with the primary trigonometric functions.

Mastering cotangent can make solving these problems much more approachable.

Difference Identities

Difference identities for trigonometric functions are incredibly useful for solving equations involving angles. They allow you to expand expressions involving the difference of two angles, such as \( \cos(x-y) \) and \( \sin(x-y) \).
For example, the difference identity for cosine:\[ \cos(x-y) = \cos x \cos y + \sin x \sin y \]
And for sine:\[ \sin(x-y) = \sin x \cos y - \cos x \sin y \]
These identities simplify the process of proving trigonometric equations. They help convert an unknown complex angle into known simpler parts.
Here's why it's so helpful:

• They transform complex trigonometric expressions in terms of known identities.
• Help in breaking down and understanding how angles relate within expressions.

By rewriting \( \cot(x-y) \) using these fundamental differences, we gain insights into solving larger trigonometric expressions.

Trigonometric Proofs

Proving trigonometric identities often requires breaking down expressions using known identities. It's a methodical process that requires following logical steps and transformations.
For example, when proving \( \cot(x-y) = \frac{\cot x \cot y + 1}{\cot y - \cot x} \), we transform each term using identities like those for sine and cosine. This involves substituting known values to simplify the original complex expression.
The process often involves:

• Recognizing and substituting identities, such as \( \cot(x) = \frac{\cos(x)}{\sin(x)} \), converting everything to sine and cosine initially.
• Following mathematical manipulations carefully, such as separating fractions and simplifying.
• Ensuring each transformation is valid and logical, not jumping steps without verification.

By practicing trigonometric proofs, students not only solidify their understanding of trigonometric functions but also enhance their problem-solving skills in mathematics.