Question

Prove each identity. (All identities in this chapter can be proven. ) $$(\sec \theta-\tan \theta)(\tan \theta+\sec \theta)=1$$

Short answer

By applying the distributive property and using trigonometric identities, the original expression simplifies to 1, thus proving the identity.

Step-by-step solution

Identify the Left-Hand Side

Start by recognizing the left-hand side of the equation as the expression we need to simplify or manipulate. In this case, it is \(\sec \theta - \tan \theta\) multiplied by \(\tan \theta + \sec \theta\).

Apply the Distributive Property

Use the distributive property to multiply the two binomials. This entails multiplying each term in the first binomial by each term in the second binomial, which gives us: \[(\sec \theta \times \tan \theta) + (\sec^2 \theta) - (\tan^2 \theta) - (\tan \theta \times \sec \theta)\].

Simplify the Expression

Notice that \(\sec \theta \times \tan \theta\) and \(\tan \theta \times \sec \theta\) are like terms but with opposite signs, so they cancel each other out. This leaves us with \(\sec^2 \theta - \tan^2 \theta\).

Use Trigonometric Identities

Recognize that \(\sec^2 \theta - \tan^2 \theta = 1\) is a well-known trigonometric identity derived from the Pythagorean identity \(\sec^2 \theta = 1 + \tan^2 \theta\). Thus, when you subtract \(\tan^2 \theta\) from both sides of the Pythagorean identity, you are left with 1.

Key concepts

Trigonometric Identities

Trigonometric identities are equations that hold true for all values within the domain of the variable, commonly an angle \theta in trigonometry. They can be thought of as the 'rules' of trigonometry and are foundational for solving problems and proving statements.

For instance, in the exercise, the identity \(\sec^2 \theta - \tan^2 \theta = 1\) is pivotal to proving the given statement. In the world of trigonometry, mastering these identities is akin to knowing the grammar of a language—essential for constructing and deconstructing mathematical 'sentences' or expressions.

Pythagorean Identity

The Pythagorean identity is one of the most fundamental concepts in trigonometry. It describes the relationship between the sine, cosine, and tangent functions of an angle based on the Pythagorean Theorem. For example, \(\sin^2 \theta + \cos^2 \theta = 1\) is the most basic Pythagorean identity.

From this, we derive the associated identity used in our exercise, \(\sec^2 \theta = 1 + \tan^2 \theta\). The Pythagorean identities are used to simplify trigonometric expressions, facilitating the process of verifying more complex trigonometric identities.

Distributive Property

The distributive property in algebra is a way to simplify expressions where one term is multiplied by a binomial or polynomial. The rule states that a term outside a parenthesis can be distributed to each term inside the parenthesis and multiplied by each independently. In formulas, it is expressed as \(a(b + c) = ab + ac\)

In our exercise, applying the distributive property to the product of the two binomials \(\sec \theta - \tan \theta\) and \(\tan \theta + \sec \theta\), we systematically expand to get a simplified form. The property comes in very handy when dealing with trigonometric expressions involving several terms.

Trigonometry

Trigonometry is a branch of mathematics that studies the relationships between angles and sides of triangles, particularly right-angled triangles. It is built upon the trigonometric functions like sine \(\sin\), cosine \(\cos\), and tangent \(\tan\), among others. These functions provide a way to abstract and navigate through angles and distances in various fields such as physics, engineering, and architecture.

The beauty of trigonometry lies in its ability to connect angles and distances, enabling the analysis of periodic phenomena and circular motion. When proving trigonometric identities like in our exercise, we rely on our knowledge of these functions and their relations to demonstrate that two expressions, which may seem unrelated at first glance, are indeed equivalent.