Problem 38 - Contemporary Precalculus

Question

Prove the addition and subtraction identities for the tangent function (page 526 ). [ Hint: $$ \tan (x+y)=\frac{\sin (x+y)}{\cos (x+y)} $$ Use the addition identities on the numerator and denominator; then divide both numerator and denominator by \(\cos x \cos y \text { and simplify. }]\)

Step-by-Step Solution

Answer

Question: Prove the addition and subtraction identities for the tangent function, given the hint to use sine and cosine addition and subtraction identities. Answer: Using the sine and cosine addition and subtraction identities, we derived the following identities for the tangent function: $$ \tan(x+y) = \frac{\tan(x) + \tan(y)}{1 - \tan(x)\tan(y)} \\ \tan(x-y) = \frac{\tan(x) - \tan(y)}{1 + \tan(x)\tan(y)} $$

Step 1: Write expressions for tangent sums and differences

Express the tangent of (x+y) and (x-y) in terms of sine and cosine functions: $$ \tan(x+y) = \frac{\sin(x+y)}{\cos(x+y)} \\ \tan(x-y) = \frac{\sin(x-y)}{\cos(x-y)} $$

Step 2: Apply sine and cosine addition and subtraction identities

Use the sine and cosine addition and subtraction identities in the numerators and denominators: $$ \tan(x+y) = \frac{\sin(x)\cos(y) + \cos(x)\sin(y)}{\cos(x)\cos(y) - \sin(x)\sin(y)} \\ \tan(x-y) = \frac{\sin(x)\cos(y) - \cos(x)\sin(y)}{\cos(x)\cos(y) + \sin(x)\sin(y)} $$

Step 3: Divide by cos(x)cos(y) and simplify

Divide both the numerator and denominator of each expression by cos(x)cos(y) and simplify the expressions: $$ \tan(x+y) = \frac{\sin(x)\cos(y) + \cos(x)\sin(y)}{\cos(x)\cos(y) - \sin(x)\sin(y)} \cdot \frac{\frac{1}{\cos(x)\cos(y)}}{\frac{1}{\cos(x)\cos(y)}} = \frac{\frac{\sin(x)}{\cos(x)}+\frac{\sin(y)}{\cos(y)}}{1-\frac{\sin(x)}{\cos(x)}\cdot \frac{\sin(y)}{\cos(y)}} $$ And similarly for \(\tan(x - y)\): $$ \tan(x-y) = \frac{\sin(x)\cos(y) - \cos(x)\sin(y)}{\cos(x)\cos(y) + \sin(x)\sin(y)} \cdot \frac{\frac{1}{\cos(x)\cos(y)}}{\frac{1}{\cos(x)\cos(y)}} = \frac{\frac{\sin(x)}{\cos(x)}-\frac{\sin(y)}{\cos(y)}}{1+\frac{\sin(x)}{\cos(x)}\cdot \frac{\sin(y)}{\cos(y)}} $$

Step 4: Recognize tangent expressions

Observe that the resulting expressions are in the form of tangent functions: $$ \tan(x+y) = \frac{\tan(x) + \tan(y)}{1 - \tan(x)\tan(y)} \\ \tan(x-y) = \frac{\tan(x) - \tan(y)}{1 + \tan(x)\tan(y)} $$ We have now proven the addition and subtraction identities for the tangent function.

Key Concepts

Tangent Function

The tangent function, often abbreviated as "tan," is a fundamental trigonometric function. It connects angles in a triangle to ratios of two specific sides. The tangent of an angle in a right-angled triangle is defined as the ratio of the opposite side to the adjacent side. This can be expressed as:\[\tan \theta = \frac{\text{Opposite Side}}{\text{Adjacent Side}}\]This function is periodic and repetitive over an interval of \(\pi\) radians, meaning its values repeat in this interval. In the context of the unit circle, the tangent function can also be expressed as the sine of the angle divided by the cosine of the angle:\[\tan \theta = \frac{\sin \theta}{\cos \theta}\]Understanding the tangent function is crucial because it helps in solving various trigonometric problems, including those involving addition and subtraction identities.

Addition Identity

The addition identity for the tangent function provides a formula for finding the tangent of the sum of two angles. Here's the fundamental idea:

• The addition formula for tangent is expressed as: \[ \tan(x+y) = \frac{\tan(x) + \tan(y)}{1 - \tan(x)\tan(y)} \]
• This formula is derived from using the addition formulas for sine and cosine functions.

To break it down:- The numerator \( \tan(x) + \tan(y) \) represents the algebraic sum of the tangents of the individual angles.- The denominator \( 1 - \tan(x)\tan(y) \) adjusts this sum by subtracting the product of the tangents of the two angles.This identity allows us to find the tangent of a combined angle using the tangents of the individual angles, which simplifies complex trigonometric problems.

Subtraction Identity

The subtraction identity for the tangent function is quite similar to its addition counterpart. It's designed to simplify the calculation of the tangent for the difference of two angles. The formula is given as:

• The subtraction formula for tangent is: \[ \tan(x-y) = \frac{\tan(x) - \tan(y)}{1 + \tan(x)\tan(y)} \]
• This formula utilizes the subtraction identities for sine and cosine functions.

In this case:- The numerator \( \tan(x) - \tan(y) \) is the difference between the tangents of the two angles.- The denominator \( 1 + \tan(x)\tan(y) \) adds the product of the two tangents to account for any overlap.This identity is incredibly useful in trigonometry as it helps solve equations involving angle differences more efficiently.

Sine and Cosine Functions

The sine and cosine functions are the foundational blocks of trigonometry. They describe the relationships between the angles and sides of right-angled triangles.

• The sine function, \( \sin \theta \), measures the ratio of the length of the side opposite the angle \( \theta \) to the hypotenuse.
• The cosine function, \( \cos \theta \), measures the ratio of the length of the adjacent side to the hypotenuse.

The addition and subtraction identities of the sine and cosine functions are used to express the sine and cosine of sums and differences of angles:- Addition Identities: \[ \sin(x+y) = \sin x \cos y + \cos x \sin y \] \[ \cos(x+y) = \cos x \cos y - \sin x \sin y \]- Subtraction Identities: \[ \sin(x-y) = \sin x \cos y - \cos x \sin y \] \[ \cos(x-y) = \cos x \cos y + \sin x \sin y \]With these identities, one can transform complex trigonometric expressions into simpler ones, making calculations easier and facilitating problem-solving in a variety of mathematical applications.