Problem 88 - 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 is verified: \( \frac{\tan x + \tan y}{\cot x + \cot y} = \tan x \tan y \).

Step 1: Rewrite Terms Using Basic Identities

Start by using the basic trigonometric identity for tangent and cotangent. We know that \( \tan x = \frac{\sin x}{\cos x} \) and \( \cot x = \frac{\cos x}{\sin x} \). Rewriting the original expression with these identities gives:\[ \frac{\frac{\sin x}{\cos x} + \frac{\sin y}{\cos y}}{\frac{\cos x}{\sin x} + \frac{\cos y}{\sin y}} \]

Step 2: Simplify the Expression

Combine the terms in the numerator and the denominator by finding a common denominator for each. For \( \tan x + \tan y \), the common denominator is \( \cos x \cos y \):\[ \frac{\sin x \cos y + \sin y \cos x}{\cos x \cos y} \]For \( \cot x + \cot y \), the common denominator is \( \sin x \sin y \):\[ \frac{\cos x \sin y + \cos y \sin x}{\sin x \sin y} \]

Step 3: Formulate the Entire Fraction

Now substitute the simplified versions of the numerator and denominator back into the fraction:\[ \frac{\frac{\sin x \cos y + \sin y \cos x}{\cos x \cos y}}{\frac{\cos x \sin y + \cos y \sin x}{\sin x \sin y}} \]

Step 4: Simplify the Compound Fractions

Since we have a fraction divided by a fraction, multiply by the reciprocal of the denominator:\[ \frac{\sin x \cos y + \sin y \cos x}{\cos x \cos y} \times \frac{\sin x \sin y}{\cos x \sin y + \cos y \sin x} \]

Step 5: Recognize a Trigonometric Identity

Notice that both \( \sin x \cos y + \sin y \cos x \) and \( \cos x \sin y + \cos y \sin x \) can be expressed using the sum-to-product identities, and each simplifies to \( \sin(x + y) \). Therefore, we have:\[ \frac{\sin(x+y)}{\cos x \cos y} \times \frac{\sin x \sin y}{\sin(x+y)} \]

Step 6: Cancel Identical Terms

Cancel the \( \sin(x+y) \) terms in the expression. This simplifies the expression to:\[ \frac{\sin x \sin y}{\cos x \cos y} \]

Step 7: Simplify Further

Recognize that \( \frac{\sin x \sin y}{\cos x \cos y} = \tan x \tan y \). Thus, we have shown\[ \frac{\tan x + \tan y}{\cot x + \cot y} = \tan x \tan y \]

Step 8: Conclusion

We have verified the identity:\[ \frac{\tan x + \tan y}{\cot x + \cot y} = \tan x \tan y \]

Key Concepts

Tangent and its Role in Trigonometry

Tangent, often abbreviated as \( \tan \), is a fundamental trigonometric function, which represents the ratio of the sine to the cosine of an angle. In formulaic terms, this can be expressed as \( \tan x = \frac{\sin x}{\cos x} \). Tangent is crucial in solving various trigonometric equations and identities due to its unique relationship within the unit circle.
This particular identity \( \frac{\tan x + \tan y}{\cot x + \cot y} = \tan x \tan y \), challenges us to dive deep into tangent's properties and how it interacts with other functions. It's important to remember:

• Tangent is periodic with a period of \( \pi \).
• The function is undefined wherever cosine equals zero, which occurs at odd multiples of \( \pi/2 \).

Moreover, understanding how tangent behaves under transformations can simplify solving such trigonometric identities.

Understanding Cotangent

Cotangent, or \( \cot \), is another core trigonometric function that is the reciprocal of tangent. Mathematically, \( \cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x} \). It's crucial to grasp cotangent's behavior, especially when involved in identities with tangent.
As with tangent, cotangent also has certain properties that aid in simplifying trigonometric expressions:

• Cotangent is periodic with a period of \( \pi \).
• It is undefined wherever sine equals zero, which occurs at integral multiples of \( \pi \).

In the given identity exercise, converting terms to their sine and cosine forms allows us to manipulate and simplify the expression effectively. This forms the basis for combining and simplifying \( \tan x + \tan y \) and \( \cot x + \cot y \) before further transformations.

Exploring Sum-to-Product Identities

Sum-to-product identities are powerful tools in trigonometry that allow the simplification of expressions involving the sum or difference of trigonometric functions. These identities are derived from the fundamental sine and cosine addition formulas.
For example, in our exercise, we observed the appearance of expressions such as \( \sin x \cos y + \sin y \cos x \). Recognizing these expressions as equivalent to \( \sin(x + y) \) through sum-to-product identities simplifies the computation drastically.

• These identities help in transforming summations such as \( \sin x + \sin y \) or \( \cos x + \cos y \) into products.
• Simplification not only reduces complexity but also unveils hidden patterns and relationships.

By applying the sum-to-product identities effectively, we can cancel out terms and reach our objective of verifying or simplifying trigonometric identities, as seen in the provided step-by-step solution.