Problem 35 - Precalculus with Calculus Previews

Question

Verify the given identity. $$ \sec t-\frac{\cos t}{1+\sin t}=\tan t $$

Step-by-Step Solution

Answer

The identity is verified: \( \sec t-\frac{\cos t}{1+\sin t}=\tan t \).

Step 1: Rewrite Secant and Cosine in Terms of Sine and Cosine

The identity involves secant and cosine. We will first rewrite secant as \( \sec t = \frac{1}{\cos t} \), hence \( \sec t - \frac{\cos t}{1 + \sin t} \) becomes \( \frac{1}{\cos t} - \frac{\cos t}{1 + \sin t} \).

Step 2: Find a Common Denominator

To simplify the expression, we need a common denominator for the two terms. The first term has a denominator of \( \cos t \) and the second has \( 1 + \sin t \). The common denominator is \( \cos t (1 + \sin t) \).

Step 3: Rewrite Each Term with the Common Denominator

Rewrite \( \frac{1}{\cos t} \) as \( \frac{1 + \sin t}{\cos t(1 + \sin t)} \) and \( \frac{\cos t}{1 + \sin t} \) as \( \frac{\cos^2 t}{\cos t (1 + \sin t)} \).

Step 4: Simplify the Expression

The expression with common denominators is: \( \frac{1 + \sin t}{\cos t(1 + \sin t)} - \frac{\cos^2 t}{\cos t(1 + \sin t)} \). Thus, the numerator becomes \( (1 + \sin t) - \cos^2 t \).

Step 5: Use the Pythagorean Identity

Recall that \( \cos^2 t = 1 - \sin^2 t \). Substitute this into the numerator: \( (1 + \sin t) - (1 - \sin^2 t) = \sin t + \sin^2 t \).

Step 6: Factor the Numerator

Factor out a sin from the numerator: \( \sin t (1 + \sin t) \). So the expression becomes \( \frac{\sin t (1 + \sin t)}{\cos t (1 + \sin t)} \).

Step 7: Cancel Common Terms

Cancel the common term \( 1 + \sin t \) in the numerator and denominator. You now have \( \frac{\sin t}{\cos t} \).

Step 8: Identity Verification

Recognize that \( \frac{\sin t}{\cos t} \) is equal to \( \tan t \). Thus, the original expression \( \sec t - \frac{\cos t}{1 + \sin t} \) is equal to \( \tan t \). This verifies the identity.

Key Concepts

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles and sides of a triangle to each other. They are foundational in trigonometry, helping to solve problems involving triangles. The primary trigonometric functions are sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)). Each of these functions takes an angle as input and relates it to a ratio of sides in a right triangle.

• Sine of an angle is the ratio of the length of the opposite side to the hypotenuse: \( \sin t = \frac{\text{Opposite}}{\text{Hypotenuse}} \).
• Cosine is the ratio of the adjacent side to the hypotenuse: \( \cos t = \frac{\text{Adjacent}}{\text{Hypotenuse}} \).
• Tangent is the ratio of the opposite side to the adjacent side: \( \tan t = \frac{\text{Opposite}}{\text{Adjacent}} \).

In addition, we have other functions derived from these, such as secant (\( \sec \)), cosecant (\( \csc \)), and cotangent (\( \cot \)). For this exercise, understanding secant is important, as \( \sec t = \frac{1}{\cos t} \), demonstrating its close relationship with cosine. Recognizing these relationships is key when simplifying or manipulating trigonometric expressions.

Pythagorean Identity

The Pythagorean identity is one of the most essential identities in trigonometry. It comes from the well-known Pythagorean theorem and provides a fundamental relationship between the sine and cosine of an angle. The identity is expressed as:
\[ \sin^2 t + \cos^2 t = 1 \]
This identity can be transformed in various ways to solve problems or simplify expressions.

• For example, you can express cosine squared in terms of sine: \( \cos^2 t = 1 - \sin^2 t \).
• Similarly, \( \sin^2 t \) can be written as \( 1 - \cos^2 t \).

In the provided solution, the Pythagorean identity is used to simplify the expression by replacing \( \cos^2 t \) with \( 1 - \sin^2 t \). This transformation is crucial for recognizing patterns within trigonometric identities and facilitating algebraic manipulation. Grasping this identity helps in simplifying complex trigonometric equations.

Factoring Algebraic Expressions

Factoring is a powerful algebraic tool used to simplify expressions and solve equations. When it comes to trigonometric expressions, factoring can reveal hidden simplifications, making it easier to manipulate and verify identities. Let’s break down the process:

• Factoring involves rewriting an expression as a product of its factors. For example, the expression \( \sin t + \sin^2 t \) can be factored by taking \( \sin t \) common, giving \( \sin t (1 + \sin t) \).
• This process not only simplifies the expression but also helps in canceling common terms in fractions, allowing for easier simplification. In our exercise, spotting the common factor \( 1 + \sin t \) allows us to eliminate it from both the numerator and denominator.

Once factored, the expression becomes more manageable, aiding in the verification of trigonometric identities. For instance, after factoring and simplifying the expression \( \sec t - \frac{\cos t}{1 + \sin t} \) to \( \frac{\sin t}{\cos t} \), recognizing it as \( \tan t \) completes the identity verification perfectly. Understanding how to factor expressions deeply enhances the ability to solve and simplify complex trigonometric problems.