Problem 34 - Algebra and Trigonometry

Question

\(23-40\) . Prove the identity. $$ \cot (x+y)=\frac{\cot x \cot y-1}{\cot x+\cot y} $$

Step-by-Step Solution

Answer

The identity is proven valid using the cotangent and tangent angle sum formulas.

Step 1: Understand the trigonometric identity

The expression given to us is an identity involving cotangent functions: \( \cot (x+y) = \frac{\cot x \cot y - 1}{\cot x + \cot y} \). The goal is to prove that the left side of the equation is indeed equal to the right side.

Step 2: Recall the cotangent angle sum formula

The formula for \( \cot (x+y) \) is given by the reciprocal of the tangent angle sum formula: \( \cot (x+y) = \frac{\cot x \cot y - 1}{\cot x + \cot y} \) can also be understood by first expressing it as \( \cot (x+y) = \frac{1}{\tan(x+y)} \). Therefore, utilize the tangent of sum formula: \( \tan(x+y) = \frac{\tan x + \tan y}{1 - \tan x \tan y} \). Hence, \( \cot(x+y) = \frac{1 - \tan x \tan y}{\tan x + \tan y} \).

Step 3: Express \(\tan x\) and \(\tan y\) in terms of \(\cot x\) and \(\cot y\)

We know \( \tan x = \frac{1}{\cot x} \) and \( \tan y = \frac{1}{\cot y} \). Substitute these into the expression for \( \tan(x+y) \) from step 2 to get: \( \tan(x+y) = \frac{\frac{1}{\cot x} + \frac{1}{\cot y}}{1 - \frac{1}{\cot x} \cdot \frac{1}{\cot y}} \).

Step 4: Simplify the expression

Replace \( \tan x \) and \( \tan y \) in the equation \( \tan(x+y) = \frac{\cot x + \cot y}{\cot x \cot y - 1} \) following derivations from previous steps. The expression for \( \tan(x+y) \) becomes: \( \tan(x+y) = \frac{\cot x \cot y - 1}{\cot x + \cot y} \). Therefore, \( \cot(x+y) = \frac{\cot x \cot y - 1}{\cot x + \cot y} \).

Step 5: Confirm identity verification

After substituting and simplifying both sides of the original identity equation, they resulted in being equal. Hence, the identity \( \cot (x+y) = \frac{\cot x \cot y - 1}{\cot x + \cot y} \) is proven as valid.

Key Concepts

Cotangent Function

The cotangent function, often represented as \( \cot \), is one of the fundamental trigonometric functions. It relates to the tangent function, providing the inverse value. Specifically, for an angle \( x \), the cotangent is the ratio of the cosine and sine of that angle:

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

This relation helps to understand various trigonometric identities and simplify complex trigonometric expressions.
The cotangent function has some intriguing properties:

• It is undefined wherever the sine of the angle is zero, leading to vertical asymptotes at those points on its graph.
• The function is periodic with a period of \( \pi \).
• In some contexts, especially involving angle sum formulae, cotangent proves to be quite handy, connecting directly to tangent through simple reciprocation.

Understanding \( \cot x \) as the reciprocal of \( \tan x \) (since \( \tan x = \frac{1}{\cot x} \)) is vital when dealing with trigonometric identities and simplifying them.

Angle Sum Formula

The angle sum formula is a crucial tool in trigonometry that allows us to determine the trigonometric function values of sums or differences of angles. Specifically, for tangent, the angle sum formula is:

• \( \tan(x+y) = \frac{\tan x + \tan y}{1 - \tan x \tan y} \)

This expression gives the tangent of the sum of two angles \( x \) and \( y \) in terms of their individual tangents. Using this formula is essential when proving identities like the one in our exercise.
The angle sum formula is especially versatile as:

• It can be rearranged and used in reverse to separate angles when needed.
• It extends to other trigonometric functions like sine and cosine as well, providing a comprehensive approach to angle manipulation.
• When using the tangent angle sum formula, understanding it in terms of \( \tan x = \frac{1}{\cot x} \) is helpful, especially when we need to switch between tangent and cotangent.

Such formulas are cornerstones in trigonometry, allowing for simplification and proof of more complex identities.

Tangent Function

The tangent function, symbolized as \( \tan \), is another primary trigonometric function. It is defined as the ratio of the sine and cosine of an angle:

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

Tangent is fundamentally linked to cotangent through its reciprocal relationship. This reciprocal nature often aids in transforming from one function to the other, particularly in proofs involving identities like those using the angle sum formula.
Some details regarding the tangent function include:

• It is undefined wherever the cosine of the angle is zero, creating vertical asymptotes in its graph.
• It has a period of \( \pi \), similar to cotangent, meaning it repeats values every \( \pi \) units.
• When tackling identities such as the one in the exercise, knowing how to express tangent in terms of cotangent helps to simplify and prove complex relationships.

In essence, mastering the tangent function and its relationship with cotangent through reciprocal identities is key to understanding broader trigonometric principles.