Problem 39 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The identity is verified by expanding and simplifying to get \(\tan y\).

Step 1: Expand the Numerator

\(\sin(x+y) - \sin(x-y)\):
\(= (\sin x\cos y + \cos x\sin y) - (\sin x\cos y - \cos x\sin y)\)
\(= 2\cos x\sin y\)

Step 2: Expand the Denominator

\(\cos(x+y) + \cos(x-y)\):
\(= (\cos x\cos y - \sin x\sin y) + (\cos x\cos y + \sin x\sin y)\)
\(= 2\cos x\cos y\)

Step 3: Simplify

\(\frac{2\cos x\sin y}{2\cos x\cos y} = \frac{\sin y}{\cos y} = \tan y\)

Key Concepts

Addition Formulas

Addition formulas are key identities in trigonometry that help us find the sine, cosine, and tangent of the sum or difference of two angles. In our problem, we use these formulas to simplify and manipulate the given expression.

The addition formulas for sine and cosine are:

• Sine: \( \sin(a + b) = \sin a \cos b + \cos a \sin b \)
• Sine (difference): \( \sin(a - b) = \sin a \cos b - \cos a \sin b \)
• Cosine: \( \cos(a + b) = \cos a \cos b - \sin a \sin b \)
• Cosine (difference): \( \cos(a - b) = \cos a \cos b + \sin a \sin b \)

These allow us to break down expressions with angles like \( x + y \) and \( x - y \) into more manageable parts.

To solve the identity, substitute these formulas into our expression. This will help us simplify the numerator and denominator separately.

Sine Function

The sine function is one of the fundamental trigonometric functions. It represents the y-coordinate of a point on the unit circle at a given angle. Understanding this function is crucial for applying the addition formulas for sine.

When we discuss the sine of angles like \( x + y \) or \( x - y \), the addition formulas come into play. For the given problem \( \sin(x+y) - \sin(x-y) \), substitute the corresponding expressions from the addition formulas:

• For \( \sin(x+y) \): \( \sin x \cos y + \cos x \sin y \)
• For \( \sin(x-y) \): \( \sin x \cos y - \cos x \sin y \)

Putting these terms together, the \( \cos x \sin y \) parts add up, and we simplify it to \( 2\cos x \sin y \). Recognizing how these terms interact simplifies the operation within trigonometric identities.

Cosine Function

The cosine function, like sine, is another key component in trigonometry. It denotes the x-coordinate of a point on the unit circle. This function is crucial in the cosine addition formulas used in our problem.

For the expression \( \cos(x+y) + \cos(x-y) \), apply the addition formulas to break it down:

• \( \cos(x+y) = \cos x \cos y - \sin x \sin y \)
• \( \cos(x-y) = \cos x \cos y + \sin x \sin y \)

When these are combined, the \( \sin x \sin y \) terms cancel out, resulting in \( 2\cos x \cos y \). This simplification is essential in evaluating the overall expression and helps in deducing the final identity.

Tangent Function

The tangent function represents the ratio of the sine and cosine functions. It's a key trigonometric function often useful in converting complex identities into simpler ones.

In our task, we start by expressing \( \tan y \) in terms of sine and cosine:

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

After simplifying both the numerator \( 2\cos x \sin y \) and the denominator \( 2\cos x \cos y \) of the expression using sine and cosine identities, we find:

• \( \frac{2\cos x \sin y}{2\cos x \cos y} = \frac{\sin y}{\cos y} \)

This confirms the identity, \( \tan y \), after cancellation. Simplifying complex expressions using basic trigonometric functions like tangent makes solving identities straightforward.