Problem 6 - Algebra 2

Question

Verify that each of the following is an identity. $$ \frac{\sec \theta+1}{\tan \theta}=\frac{\tan \theta}{\sec \theta-1} $$

Step-by-Step Solution

Answer

The given equation is an identity, as both sides reduce to \(\sin^2 \theta\).

Step 1: Simplify the left-hand side (LHS)

Start with the left-hand side of the equation: \( \frac{\sec \theta + 1}{\tan \theta} \). Recall the trigonometric identities: \( \sec \theta = \frac{1}{\cos \theta} \) and \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).Substitute these identities in:\[\frac{\sec \theta + 1}{\tan \theta} = \frac{\frac{1}{\cos \theta} + 1}{\frac{\sin \theta}{\cos \theta}} = \frac{\frac{1 + \cos \theta}{\cos \theta}}{\frac{\sin \theta}{\cos \theta}} \]This expression simplifies to:\[\frac{1 + \cos \theta}{\sin \theta}\]

Step 2: Simplify the right-hand side (RHS)

Now, simplify the right-hand side of the equation: \( \frac{\tan \theta}{\sec \theta - 1} \). Again, use the identities: \( \sec \theta = \frac{1}{\cos \theta} \) and \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).Substitute these identities in:\[\frac{\tan \theta}{\sec \theta - 1} = \frac{\frac{\sin \theta}{\cos \theta}}{\frac{1}{\cos \theta} - 1} = \frac{\sin \theta}{\frac{1 - \cos \theta}{\cos \theta}}\]This expression simplifies to:\[\frac{\sin \theta \cdot \cos \theta}{1 - \cos \theta}\]

Step 3: Verify the identity

To verify if both sides are equal, we equate the simplified expressions from Steps 1 and 2.We have:\[\frac{1 + \cos \theta}{\sin \theta} \quad \text{and} \quad \frac{\sin \theta \cdot \cos \theta}{1 - \cos \theta}\]Multiply both sides by \((1 - \cos \theta)\) and \(\sin \theta\) to clear the fractions:\[(1 + \cos \theta)(1 - \cos \theta) = \sin^2 \theta\]This is an identity because \(1 - \cos^2 \theta = \sin^2 \theta\), which confirms that both sides are indeed equal.

Key Concepts

Secant

Secant, often denoted as \( \sec \theta \), is one of the six fundamental trigonometric functions. It is defined as the reciprocal of the cosine function.

To put it simply, secant finds its value by dividing 1 by the cosine of the angle \( \theta \). Mathematically, it can be expressed as:

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

The secant function is essential in trigonometry as it provides an easy way to work with reciprocals of the cosine values, which is useful in many mathematical proofs and calculations.

In the original exercise, secant plays a key role in translating the given identity into terms of sine and cosine. This simplification process allows us to verify the identity using well-known trigonometric relationships. Understanding secant and its connection to cosine helps students easily identify how to manipulate and simplify expressions using trigonometric identities.

Tangent

Tangent, represented as \( \tan \theta \), is another primary trigonometric function. Tangent relates to the ratio of two other trigonometric functions: sine and cosine.

This is how it is typically defined:

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

The tangent function measures the steepness or slope of a line related to the unit circle and angle \( \theta \). It becomes infinitely large at certain points, leading to its undefined nature at angles where cosine equals zero.

In the exercise, tangent is transformed into its sine and cosine components to help simplify both sides of the equation. By doing so, we can utilize direct comparisons between left and right sides, ultimately proving or disproving the identity in question. This method leverages the trigonometric identity that calculates tangent as a ratio between sine and cosine, making it a versatile tool in problem-solving.

Cosine

Cosine, notated as \( \cos \theta \), is a fundamental component of trigonometry and is closely associated with the adjacent side and hypotenuse in a right-angled triangle. In the context of the unit circle, cosine represents the x-coordinate of a point on the circumference as the angle varies.

The definition of cosine can be remembered as:

• In a right triangle: \( \cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} \)
• In a unit circle: As the x-coordinate at a given angle \( \theta \)

In the process of simplifying trigonometric expressions or proving identities, cosine often serves as a reference point since it is directly linked to secant. In our exercise, converting all terms to their sine and cosine equivalents helps simplify expressions significantly. This technique capitalizes on cosine's supportive role in calculating secant and tangent, streamline verification processes, and uses the identity \( \cos^2 \theta + \sin^2 \theta = 1 \) to equate terms and verify equations.

Sine

Sine, represented as \( \sin \theta \), is one of the primary trigonometric functions alongside cosine and tangent. It is crucial for describing the properties of waves and circular motion, making it fundamental for solving various geometric and mathematical problems.

The sine function is typically described by:

• In a right triangle: \( \sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} \)
• In a unit circle: As the y-coordinate corresponding to a point defined by the angle \( \theta \)

In validating trigonometric identities, the sine function often pairs with cosine due to their combined role in the Pythagorean identity: \( \sin^2 \theta + \cos^2 \theta = 1 \). For the given problem, sine is important for re-expressing tangent, as it simplifies the problem solving by allowing us to work with uniformly defined functions. Understanding how sine and cosine interrelate through the unit circle or right triangles can facilitate the verification of complex trigonometric expressions difficult to intuitively understand otherwise.