Problem 21 - Algebra 2 and Trigonometry

Question

In \(3-26,\) prove that each equation is an identity. $$ \frac{\tan ^{2} \theta}{\sec \theta-1}-1=\sec \theta $$

Step-by-Step Solution

Answer

The equation is an identity as both sides simplify to \( \sec \theta \).

Step 1: Analyze the given equation

The given equation is \( \frac{\tan ^{2} \theta}{\sec \theta-1}-1=\sec \theta \). Our task is to show that this equation holds true for all values of \( \theta \) where both sides are defined.

Step 2: Simplify the expression involving \( \tan \theta \) and \( \sec \theta \)

First, remember the identity \( \tan^2 \theta = \sec^2 \theta - 1 \). Substitute this into the equation:\[ \frac{\sec^2 \theta - 1}{\sec \theta - 1} - 1 = \sec \theta \]

Step 3: Simplify the left-hand side

Start by rewriting the fraction:\[ \frac{(\sec^2 \theta - 1)}{(\sec \theta - 1)} \]Observe that the numerator can be factored using the difference of squares formula: \( (\sec \theta - 1)(\sec \theta + 1) \). Substitute back:\[ \frac{(\sec \theta - 1)(\sec \theta + 1)}{\sec \theta - 1} \]

Step 4: Cancel terms

Since \( \sec \theta eq 1 \), we can simplify \( \frac{(\sec \theta - 1)(\sec \theta + 1)}{\sec \theta - 1} = \sec \theta + 1 \).Thus, the expression becomes:\[ \sec \theta + 1 - 1 \]

Step 5: Further simplify the left-hand side

The expression \( \sec \theta + 1 - 1 \) simplifies to \( \sec \theta \). This shows that the left-hand side simplifies to the right-hand side:\[ \sec \theta = \sec \theta \]

Step 6: Conclusion

Both sides of the given equation are equal after simplification, proving that the original equation is an identity.

Key Concepts

Tangent and Secant functions

In trigonometry, the tangent and secant functions are closely related to the right triangle and the unit circle. Understanding these functions requires a good grasp of their definitions and how they connect to other trigonometric identities.
The tangent function, denoted as \( \tan \theta \), is the ratio of the opposite side to the adjacent side in a right triangle. It can also be defined using sine and cosine functions:

• \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

The secant function, represented as \( \sec \theta \), is the reciprocal of the cosine function:

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

These definitions are vital for simplifying trigonometric expressions. By expressing \( \tan^2 \theta \) as \( \sec^2 \theta - 1 \), we establish a relationship that assists in understanding their behavior in equations and identities.

Applying such identities correctly is key to transforming complex trigonometrical expressions into solvable forms.

Trigonometric Simplification

Simplifying trigonometric expressions is a fundamental skill in trigonometry. It's about transforming complicated expressions into simpler, more manageable forms. Here, we focus on using identities to make this task more straightforward.
The original problem involves simplifying \( \frac{\tan^2 \theta}{\sec \theta - 1} - 1 \). The goal is to express components in terms of a single trigonometric function, like \( \sec \theta \), using known identities. One powerful tool is the Pythagorean identity:

• \( \tan^2 \theta = \sec^2 \theta - 1 \)

Substituting this into the equation simplifies it. By rewriting complex fractions and breaking down terms using identities, trigonometric problems become easier to solve.
Employing methods like factorization, applying known identities repeatedly, and recognizing patterns is crucial for simplification. They not only make calculations more straightforward but also help in understanding the inherent symmetry and relationships in trigonometric functions.

Difference of Squares

The difference of squares is a handy algebraic tool often used in simplifying expressions. It takes the form:

• \( a^2 - b^2 = (a - b)(a + b) \)

This formula is useful in trigonometry, especially when dealing with expressions involving squares of trigonometric functions.
In the given exercise, the numerator in the expression \( \frac{\sec^2 \theta - 1}{\sec \theta - 1} \) can be defined as the difference of squares \((\sec \theta)^2 - (1)^2\). By recognizing this pattern, it simplifies to:

• \((\sec \theta - 1)(\sec \theta + 1)\)

When this is divided by \( \sec \theta - 1 \), it allows straightforward cancellation of terms, confirming that the expression is indeed simplified.
Understanding and recognizing the difference of squares in trigonometric identities can be a significant asset when solving problems. It makes complex algebra appear more manageable and helps unveil the simplicity inherent in many trigonometric equations.