Problem 46 - Precalculus with Calculus Previews

Question

Verify the given identity. $$ \frac{\tan (-t)}{\sin (-t)}=\sec t $$

Step-by-Step Solution

Answer

The identity is verified; \(\frac{\tan(-t)}{\sin(-t)} = \sec(t)\) simplifies correctly to \(\sec(t)\).

Step 1: Understand the Trigonometric Identities Involved

The problem involves various trigonometric identities. We need to recall some fundamental identities related to tangent, sine, and secant functions:1. \( \tan(-t) = -\tan(t) \) (odd function property of tangent).2. \( \sin(-t) = -\sin(t) \) (odd function property of sine).3. \( \sec(t) = \frac{1}{\cos(t)} \) (definition of secant in terms of cosine). These will help us transform and simplify the given expression.

Step 2: Simplify the Left Side of the Equation

Let's start with the left side of the given identity: \( \frac{\tan (-t)}{\sin (-t)} \). Using the identities from Step 1, we substitute:\[ \frac{\tan (-t)}{\sin (-t)} = \frac{-\tan(t)}{-\sin(t)} = \frac{\tan(t)}{\sin(t)}.\] Here, the negatives cancel each other out.

Step 3: Relate Tan and Sin to Cosine

We know that \( \tan(t) = \frac{\sin(t)}{\cos(t)} \). Using this relation, substitute into the expression we have:\[ \frac{\tan(t)}{\sin(t)} = \frac{\frac{\sin(t)}{\cos(t)}}{\sin(t)}.\] This simplifies further to:\[ \frac{1}{\cos(t)}.\]

Step 4: Identify Equivalent Expressions

From Step 3, \( \frac{1}{\cos(t)} \) is exactly the same as \( \sec(t) \). Therefore, we have successfully shown that the left side of the given identity simplifies to: \( \sec(t) \), which matches the right side of the equation.

Step 5: Verify the Simplified Identity

Both sides of the original equation \( \frac{\tan(-t)}{\sin(-t)} = \sec(t) \) were simplified down to \( \sec(t) \). Thus, the identity holds true.

Key Concepts

Odd Function Property

In trigonometry, understanding the odd function property is crucial. This property tells us how certain trigonometric functions behave with negative inputs, making it easier to manipulate and simplify expressions.

For example, an odd function has the property that \( f(-x) = -f(x) \). This means that if you input a negative value into the function, the output is the negative of the function's value at the positive input.

Common odd trigonometric functions include:

• Sine function: \( \sin(-t) = -\sin(t) \)
• Tangent function: \( \ an(-t) = -\tan(t) \)

Applying this knowledge, we can tackle trigonometric identities more easily. For instance, in an expression like \( \frac{\tan(-t)}{\sin(-t)} \), you can use the odd property to rewrite it as \( \frac{-\tan(t)}{-\sin(t)} \) allowing these negatives to cancel out.

This step is instrumental in reducing complexity and in the process of verifying identities as it helps transform the expression into a more recognizable form.

Secant Function

The secant function, noted by \( \sec(t) \), is one of the six fundamental trigonometric functions. It's often used in various trigonometric identities and simplifies into other functions.

The secant function is defined as the reciprocal of the cosine function, written as: \[ \sec(t) = \frac{1}{\cos(t)} \].

This relationship is pivotal for simplifying expressions. Whenever you encounter \( \sec(t) \), you know it can be replaced with \( \frac{1}{\cos(t)} \).

Understanding this simple substitution allows for simplified manipulation of trigonometric expressions and is particularly handy in calculus and algebra, where multiple transformations and simplifications take place.

In verifying identities such as \( \frac{\tan(-t)}{\sin(-t)} = \sec(t) \), using the form \( \frac{1}{\cos(t)} \) can help dissect and simplify the process by breaking down the equation into known identities, making the verification straightforward and clear.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions involves using identities and properties to rewrite expressions in simpler forms, which is essential in solving equations and verifying identities.

Start by identifying the functions involved: sine, cosine, tangent, secant, etc. Recognize the relationships and properties like odd functions or reciprocal identities.

• For tangent \( \tan(t) \): Use \( \tan(t) = \frac{\sin(t)}{\cos(t)} \) to break it into sine and cosine.
• For secant \( \sec(t) \): Recall it's \( \frac{1}{\cos(t)} \) to quickly substitute.
• Apply the odd function property: Convert expressions like \( \tan(-t) \) and \( \sin(-t) \) using their definitions.

After substitutions and using identities, look for opportunities to cancel out terms or rewrite fractions in simpler forms.

In the problem of ensuring \( \frac{\tan(-t)}{\sin(-t)} = \sec(t) \), using properties and relations effectively lets you realize that it equates to \( \sec(t) \), confirming the identity. This technique is a powerful tool in your mathematics toolkit, helping you solve and verify identities with greater ease.