Problem 89 - Precalculus Mathematics for Calculus

Question

Prove the identity. $$\frac{\sin x+\sin y}{\cos x+\cos y}=\tan \left(\frac{x+y}{2}\right)$$

Step-by-Step Solution

Answer

The identity is proven by using sum-to-product identities and simplifying.

Step 1: Reformulate the Right Side

We start by rewriting \( \tan \left( \frac{x+y}{2} \right) \) in terms of sine and cosine for better comparison. Recall that \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), so \( \tan \left( \frac{x+y}{2} \right) = \frac{\sin \left( \frac{x+y}{2} \right)}{\cos \left( \frac{x+y}{2} \right)} \).

Step 2: Use Sum-to-Product Identities

Utilize the sum-to-product identities: \( \sin x + \sin y = 2 \sin \left( \frac{x+y}{2} \right) \cos \left( \frac{x-y}{2} \right) \) and \( \cos x + \cos y = 2 \cos \left( \frac{x+y}{2} \right) \cos \left( \frac{x-y}{2} \right) \).

Step 3: Substitute Identities into The Original Expression

Substitute the identities from Step 2 into the original expression: \( \frac{2 \sin \left( \frac{x+y}{2} \right) \cos \left( \frac{x-y}{2} \right)}{2 \cos \left( \frac{x+y}{2} \right) \cos \left( \frac{x-y}{2} \right)} \).

Step 4: Simplify the Expression

The \( 2 \cos \left( \frac{x-y}{2} \right) \) terms can be cancelled from the numerator and the denominator, leaving \( \frac{\sin \left( \frac{x+y}{2} \right)}{\cos \left( \frac{x+y}{2} \right)} \).

Step 5: Recognize the Definition of Tangent

The simplified expression, \( \frac{\sin \left( \frac{x+y}{2} \right)}{\cos \left( \frac{x+y}{2} \right)} \), is precisely \( \tan \left( \frac{x+y}{2} \right) \), thus proving the identity.

Key Concepts

Sum-to-Product Identities

Sum-to-product identities help us transform sums of trigonometric functions into products. This is particularly useful when simplifying expressions or solving equations. The main identities you should know include:

• the sum of sines: \(\sin A + \sin B = 2 \sin \left( \frac{A+B}{2} \right) \cos \left( \frac{A-B}{2} \right)\)
• the sum of cosines:\(\cos A + \cos B = 2 \cos \left( \frac{A+B}{2} \right) \cos \left( \frac{A-B}{2} \right) \)

When using these formulas, understand they help "compress" the expression into a form that is often easier to manipulate or compare to another expression. In the given exercise, these identities are key to transforming the original expression into one involving the tangent function.

Trigonometric Functions

Trigonometric functions are the backbone of trigonometry, and they help describe the relationships in triangles and periodic phenomena. The basic trigonometric functions include sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)). You might also encounter their reciprocals: cosecant (\(\csc\)), secant (\(\sec\)), and cotangent (\(\cot\)).Key relationships include the Pythagorean identity:

• \(\sin^2 \theta + \cos^2 \theta = 1\)
• \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)

Understanding these functions and their relationships is vital. In our exercise, converting the tangent of a sum directly helps in simplifying and proving trigonometric identities.

Simplification in Trigonometry

Simplification in trigonometry involves rewriting complex trigonometric expressions into simpler or more manageable forms. This can involve several techniques:

• Substituting identities: Replace parts of an expression using known identities like the Pythagorean or sum-to-product identities.
• Canceling terms: Sometimes expressions include terms in both the numerator and the denominator that can be canceled.
• Converting expressions: Converting difficult functions into equivalent forms (like tan to sine and cosine) can aid in comparison or simplification.

Through these techniques, expressions become easier to work with, as seen in our exercise where the use of sum-to-product identities and conversion to basic functions helps remove redundancies and simplifies to the tangent function. This makes the mathematical argument clearer and the solution more straightforward.