Problem 32 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find identities analogous to the addition identities for each expression. (a) \(\sin (x-y)\) (b) \(\cos (x-y)\) (c) \(\tan (x-y)\)

Step-by-Step Solution

Answer

(a) \( \sin(x-y) = \sin x \cos y - \cos x \sin y \), (b) \( \cos(x-y) = \cos x \cos y + \sin x \sin y \), (c) \( \tan(x-y) = \frac{\tan x - \tan y}{1 + \tan x \tan y} \).

Step 1: Recall the Sine Addition Formula

The sine addition formula is \( \sin(a + b) = \sin a \cos b + \cos a \sin b \). We will modify this formula to find the identity for \( \sin(x - y) \).

Step 2: Apply the Sine Subtraction Formula

To find the expression for \( \sin(x-y) \), replace \( b \) with \( -y \) in the sine addition formula: \( \sin(x - y) = \sin x \cos(-y) + \cos x \sin(-y) \).

Step 3: Use Trigonometric Properties of Negative Angles

Recall that \( \cos(-y) = \cos y \) and \( \sin(-y) = -\sin y \). Apply these identities to the sine formula: \( \sin(x-y) = \sin x \cos y - \cos x \sin y \).

Step 4: Recall the Cosine Addition Formula

The cosine addition formula is \( \cos(a + b) = \cos a \cos b - \sin a \sin b \). We will modify this formula to find the identity for \( \cos(x - y) \).

Step 5: Apply the Cosine Subtraction Formula

To find the expression for \( \cos(x-y) \), replace \( b \) with \( -y \) in the cosine addition formula: \( \cos(x - y) = \cos x \cos(-y) - \sin x \sin(-y) \).

Step 6: Use Trigonometric Properties of Negative Angles

Using \( \cos(-y) = \cos y \) and \( \sin(-y) = -\sin y \), apply these identities to the cosine formula: \( \cos(x-y) = \cos x \cos y + \sin x \sin y \).

Step 7: Recall the Tangent Addition Formula

The tangent addition formula is \( \tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} \). We will modify this formula to find the identity for \( \tan(x - y) \).

Step 8: Apply the Tangent Subtraction Formula

To find the expression for \( \tan(x-y) \), replace \( b \) with \( -y \) in the tangent addition formula: \( \tan(x - y) = \frac{\tan x - \tan y}{1 + \tan x \tan y} \).

Step 9: Verify the Negative Angle Property for Tangent

Using the fact that tangent is an odd function, \( \tan(-y) = -\tan y \), the formula remains valid: \( \tan(x-y) = \frac{\tan x - \tan y}{1 + \tan x \tan y} \).

Key Concepts

Sine Subtraction Formula

Understanding the Sine Subtraction Formula will help us simplify trigonometric expressions involving subtraction of angles. This formula is derived by modifying the Sine Addition Formula.
To express \(\sin(x-y)\), consider the Sine Addition Formula: \(\sin(a + b) = \sin a \cos b + \cos a \sin b\). We replace \(b\) with \(-y\), resulting in:

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

Next, use trigonometric properties of negative angles:

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

Substituting these properties yields the Sine Subtraction Formula:

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

Notice how the terms change signs in the formula. This is a key feature of the subtraction identity, emphasizing the importance of angle orientation in trigonometry.

Cosine Subtraction Formula

The Cosine Subtraction Formula is essential for transforming expressions when angles are subtracted. Similar to our approach with the sine, we'll tweak the Cosine Addition Formula.
Begin with the Cosine Addition Formula: \(\cos(a + b) = \cos a \cos b - \sin a \sin b\). To tackle \(\cos(x - y)\), substitute \(b\) with \(-y\):

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

Utilize the properties of negative angles:

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

Replacing these into the equation, we arrive at the Cosine Subtraction Formula:

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

In this formula, notice the term \(-\sin x \sin(-y)\) becomes \(+\sin x \sin y\). This showcases how subtraction can lead to addition based on how the angles are manipulated.

Tangent Subtraction Formula

The Tangent Subtraction Formula is slightly more complex compared to its sine and cosine counterparts, thanks to the nature of tangent. By modifying the Tangent Addition Formula, we can handle the subtraction of angles effectively.
Start with the formula for the addition of tangents: \(\tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b}\). To derive the expression for \(\tan(x - y)\), substitute \(b\) with \(-y\):

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

Due to tangent being an odd function, \(\tan(-y) = -\tan y\):

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

This result shows how the subtraction influences the numerator and denominator, reversing signs in the formula's components. Understanding these transformations is crucial for effectively applying the tangent identity in various settings.