Problem 39 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 37-46: Verify the identity. $$ \tan \left(u+\frac{\pi}{4}\right)=\frac{1+\tan u}{1-\tan u} $$

Step-by-Step Solution

Answer

The identity is verified: \(\tan\left(u + \frac{\pi}{4}\right) = \frac{1 + \tan u}{1 - \tan u}\).

Step 1: Understanding the Tangent Addition Formula

The problem asks us to verify the identity \(\tan\left(u + \frac{\pi}{4}\right) = \frac{1 + \tan u}{1 - \tan u}\). We start by recalling the tangent addition formula: \(\tan(a+b) = \frac{\tan a + \tan b}{1 - \tan a \tan b}\). In our case, \(a = u\) and \(b = \frac{\pi}{4}\).

Step 2: Apply the Tangent Addition Formula

Applying the addition formula, we have \[\tan\left(u + \frac{\pi}{4}\right) = \frac{\tan u + \tan\frac{\pi}{4}}{1 - \tan u \tan\frac{\pi}{4}}.\]Since \(\tan\frac{\pi}{4} = 1\), substitute this into the equation:\[\tan\left(u + \frac{\pi}{4}\right) = \frac{\tan u + 1}{1 - \tan u \cdot 1}\].

Step 3: Simplify the Expression

The expression simplifies to:\[\tan\left(u + \frac{\pi}{4}\right) = \frac{\tan u + 1}{1 - \tan u}\].This matches exactly with the given identity, \(\frac{1 + \tan u}{1 - \tan u}\).

Step 4: Conclude the Verification

Since we have confirmed that \(\tan\left(u + \frac{\pi}{4}\right)\) simplifies to \(\frac{1 + \tan u}{1 - \tan u}\), we have verified that the identity is correct.

Key Concepts

Tangent Addition Formula

The tangent addition formula is a key concept in trigonometry that helps us find the tangent of the sum of two angles. Specifically, if we want to find \(\tan(a + b)\), we use the formula: \[\tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b}.\] This formula is very handy when you know the tangents of two angles and want to determine the tangent of their sum.
In our exercise, the angles are \(a = u\) and \(b = \frac{\pi}{4}\). The formula allows us to decompose the tangent of this sum into a formula that involves simple operations—addition and subtraction of known tangents.
By applying this formula, you not only get to the required identity but also understand how trigonometric identities are derived and verified.

Verifying Identities

Verifying a trigonometric identity involves showing that two expressions are equivalent for all values for which both sides are defined. This concept is central in mathematics and is particularly important in fields dealing with functions and transformations.
To verify an identity, you often employ algebraic manipulation and known formulas, like the tangent addition formula. Using these tools, you transform one side of the equation to match the other. This process solidifies your understanding of how different trigonometric formulas and identities correlate.

• Start by rewriting and simplifying the expressions using known identities.
• Check each step to ensure the calculations are correct.
• Use symmetry and complementary angles of trigonometric functions.

Practicing these steps helps you efficiently verify identities and appreciate the flexibility and power of trigonometry.

Trigonometry

Trigonometry is a branch of mathematics that studies the relationships between the sides and angles of triangles. Going beyond just triangles, it is essential in analyzing periodic phenomena such as sound and light waves, and it forms the basis for complex mathematical concepts.
The three primary functions in trigonometry are sine, cosine, and tangent. They help relate the angles of a triangle to the ratios of its sides.
These functions extend beyond geometric applications. They’re instrumental in calculus, physics, engineering, and even in understanding day-to-day events such as navigation and architecture.

• Sine and cosine functions are periodic and are the foundation of wave equations.
• Tangent deals with angles and slopes, a concept used in real-world applications like navigation.

By exploring exercises that involve verifying identities, students can deepen their comprehension of how trigonometric functions work and are represented.