Problem 86 - Algebra and Trigonometry

Question

Verify the identity. $$ \frac{\tan x+\tan y}{\cot x+\cot y}=\tan x \tan y $$

Step-by-Step Solution

Answer

The identity \( \frac{\tan x+\tan y}{\cot x+\cot y} = \tan x \tan y \) is verified.

Step 1: Rewrite in simpler terms

The given identity is \( \frac{\tan x + \tan y}{\cot x + \cot y} = \tan x \tan y \). Start by using the fact that \( \cot x = \frac{1}{\tan x} \) and \( \cot y = \frac{1}{\tan y} \). Substitute these into the equation:\[\frac{\tan x + \tan y}{\frac{1}{\tan x} + \frac{1}{\tan y}}.\]

Step 2: Simplify the denominator

To simplify the denominator, combine the fractions:\[\frac{1}{\tan x} + \frac{1}{\tan y} = \frac{\tan y + \tan x}{\tan x \tan y}.\]Hence, the equation becomes:\[\frac{\tan x + \tan y}{\frac{\tan x + \tan y}{\tan x \tan y}}.\]

Step 3: Simplify the entire fraction

Divide the fractions by multiplying by the reciprocal:\[(\tan x + \tan y) \times \frac{\tan x \tan y}{\tan x + \tan y}.\]The \( \tan x + \tan y \) cancels out, leaving:\[\tan x \tan y.\]

Step 4: Confirm the identity

You are left with the expression \( \tan x \tan y \) on both sides of the equation. Therefore, the identity is confirmed to be true:\[\frac{\tan x + \tan y}{\cot x + \cot y} = \tan x \tan y.\]

Key Concepts

Understanding the Tangent Function

The tangent function, often abbreviated as "tan," is one of the primary trigonometric functions. It relates an angle in a right triangle to the ratio of the length of the opposite side to the adjacent side. \[\tan{\theta} = \frac{\text{opposite}}{\text{adjacent}}\]In the unit circle, the tangent of an angle is the length of the line segment that extends from the origin to the line perpendicular to the radius at that angle. It is essential in trigonometry because it helps link angles with side ratios. A few key points about the tangent function:

• The tangent function is periodic with a period of \(\pi\) radians or 180 degrees.
• Tangent is undefined at \(90^\circ\) and \(270^\circ\), where the cosine of the angle is 0, because division by zero is undefined.
• As you move through different quadrants of the unit circle, the tangent function can be positive or negative, reflecting the signs of sine and cosine.

Exploring the Cotangent Function

The cotangent function, abbreviated as "cot," is the reciprocal of the tangent function. It is defined as:\[\cot{\theta} = \frac{1}{\tan{\theta}} = \frac{\text{adjacent}}{\text{opposite}}\]This function represents the ratio of the adjacent side to the opposite side in a right triangle, flipping the relationship of the tangent.Important characteristics of the cotangent function include:

• Like tangent, the cotangent function is also periodic, but with a period of \(\pi\) radians or 180 degrees.
• It is undefined when the sine of the angle is zero (e.g., \(0^\circ\), \(180^\circ\)), as division by zero is not possible.
• Cotangent changes signs in different quadrants, reflecting the reciprocal relationship with tangent.

Understanding cotangent is crucial for solving trigonometric identities and simplifying expressions, as seen in the given problem.

Mastering Fraction Simplification

Fraction simplification is an essential skill in mathematics, helping to make expressions easier to work with. Simplifying fractions involves rewriting a fraction in its most reduced or simplest form. Consider this when simplifying:

• Identifying common factors: Look for numbers or expressions that are common to both the numerator and the denominator.
• Canceling common terms: If possible, divide both the numerator and the denominator by their greatest common factor (GCF).
• Using reciprocals: When you have a complex fraction, simplifying it often involves multiplying by the reciprocal of the denominator.

In the trigonometric identity we verified, fraction simplification played a key role.

• The initial problem involved simplifying a fraction made of trigonometric functions, specifically combining the cotangent functions as their reciprocal tangent forms.
• By applying the concept of multiplying by the reciprocal, the expression became much easier to manage and solve.
• This process helped confirm that the original equation was indeed a valid identity.