Problem 49 - Precalculus: Functions and Graphs

Question

Verify the identity by transforming the lefthand side into the right-hand side. $$\cos \theta \sec \theta=1$$

Step-by-Step Solution

Answer

The identity is verified as \( \cos \theta \sec \theta = 1 \).

Step 1: Understand the terms

The equation \( \cos \theta \sec \theta = 1 \) involves trigonometric functions. Remember that \( \sec \theta \) is the reciprocal of \( \cos \theta \), so \( \sec \theta = \frac{1}{\cos \theta} \).

Step 2: Substitute \( \sec \theta \)

Replace \( \sec \theta \) with its definition in terms of \( \cos \theta \):\[ \cos \theta \sec \theta = \cos \theta \cdot \frac{1}{\cos \theta} \]

Step 3: Simplify the expression

Simplify the expression by performing the multiplication:\[ \cos \theta \cdot \frac{1}{\cos \theta} = 1 \]

Step 4: Verify the identity

The left-hand side \( \cos \theta \sec \theta \) simplifies to 1, which matches the right-hand side of the equation. Therefore, the identity is verified.

Key Concepts

Reciprocal Identities

In trigonometry, reciprocal identities are a set of important relationships that connect trigonometric functions and their reciprocals. These identities help us understand and manipulate the relationships between different trigonometric functions more easily.
For example, the reciprocal of the cosine function (cos) is the secant function (sec). This can be expressed as the identity:

• \( \sec \theta = \frac{1}{\cos \theta} \)

Similarly, the sine function (sin) and cosecant function (csc) are reciprocals:

• \( \csc \theta = \frac{1}{\sin \theta} \)

Tangent (tan) and cotangent (cot) follow the same pattern:

• \( \cot \theta = \frac{1}{\tan \theta} \)

These reciprocal identities are incredibly useful for solving equations and verifying trigonometric identities, as they allow us to express one trigonometric function in terms of another. This was demonstrated in the exercise by rewriting \( \sec \theta \) as \( \frac{1}{\cos \theta} \).

Trigonometric Functions

Trigonometric functions are fundamental mathematical functions that relate the angles of a triangle to the lengths of its sides. They are widely used in various fields such as geometry, physics, and engineering.
There are six basic trigonometric functions:

• Cosine (\( \cos \))
• Sine (\( \sin \))
• Tangent (\( \tan \))
• Secant (\( \sec \))
• Cosecant (\( \csc \))
• Cotangent (\( \cot \))

The cosine function, for instance, is defined as the ratio of the adjacent side to the hypotenuse in a right-angled triangle. Similarly, sine is the ratio of the opposite side to the hypotenuse, and tangent is the ratio of the opposite side to the adjacent side.
Each of these functions has a unique relationship with a specific angle in the triangle, making them very powerful tools for solving problems involving angles and distances.

Simplifying Expressions

Simplifying expressions involves reducing a complex trigonometric equation into its simplest form. This process makes it easier to understand and solve equations or verify identities.
In the exercise, we simplified the expression \( \cos \theta \sec \theta \) by substituting \( \sec \theta \) with \( \frac{1}{\cos \theta} \). This turned the equation into a straightforward multiplication problem:

• \( \cos \theta \cdot \frac{1}{\cos \theta} = 1 \)

This simplification shows how reciprocal identities can help reduce a complex expression to a simple form, directly matching the right-hand side of the identity.
By breaking down the expression into its basic components and using known identities, we can often find that what initially seems complex becomes surprisingly simple. Simplifying expressions is an invaluable skill in math, helping students to verify identities and solve trigonometric equations with ease.