Problem 45 - Precalculus with Calculus Previews

Question

Verify the given identity. $$ \cos (-t) \csc (-t)=-\cot t $$

Step-by-Step Solution

Answer

The identity is verified as true: \( \cos(-t) \csc(-t) = -\cot t \).

Step 1: Understand the identity

The given identity to verify is \( \cos(-t) \csc(-t) = -\cot t \). We need to verify whether both sides of the equation are equal.

Step 2: Recall trigonometric identities for negatives

Using trigonometric identities: \( \cos(-t) = \cos t \), and \( \csc(-t) = -\csc t \). This is because cosine is an even function, while cosecant is an odd function.

Step 3: Substitute and simplify left side

Substitute the identities from Step 2 into the left side of the equation: \( \cos(-t) \csc(-t) = \cos t (-\csc t) = -\cos t \csc t \).Thus, the left side simplifies to \(-\cos t \csc t \).

Step 4: Simplify right side

The right side of the equation is \(-\cot t = -\frac{\cos t}{\sin t}\).

Step 5: Verify equality

Notice that \(\csc t = \frac{1}{\sin t}\). Hence, \(-\cos t \csc t = -\cos t \times \frac{1}{\sin t} = -\frac{\cos t}{\sin t} \), which is exactly equal to the right side, \(-\cot t\).

Step 6: Conclusion

Since both sides simplify to \(-\frac{\cos t}{\sin t}\), the identity \( \cos(-t) \csc(-t) = -\cot t \) is verified as true.

Key Concepts

Understanding Even and Odd Functions

In trigonometry, determining whether a function is even or odd helps in simplifying expressions, especially when dealing with negative angles. An even function is symmetrical about the y-axis, and for trigonometric functions like cosine, this means that \( \cos(-t) = \cos t \). This means that the cosine function doesn't change its value when the input angle is negated.
On the other hand, odd functions are symmetrical about the origin. For functions like sine or tangent, which include sine as part of their identity, this means that \( \sin(-t) = -\sin t \) and \( \tan(-t) = -\tan t \). In the case of cosecant, which is the reciprocal of sine, it is also odd, leading us to \( \csc(-t) = -\csc t \).
Recognizing these properties of trigonometric functions is crucial when simplifying identities, as it helps determine how the function value changes with negative inputs and impacts our overall simplification and verification process.

Verifying Trigonometric Identities

Verifying trigonometric identities is a process of proving that two trigonometric expressions are equivalent. This process involves transforming one side of the equation to look like the other. Our goal is to either manipulate the left side to match the right side, or vice versa.
To successfully verify an identity, it’s important to have a solid grasp of:

• Basic trigonometric definitions and relationships
• Pythagorean identities, such as \( \sin^2 t + \cos^2 t = 1 \)
• Reciprocal identities, such as \( \csc t = \frac{1}{\sin t} \)
• Even and odd identities, like those we used earlier

Once you understand these principles, you can apply operations like factoring, adding fractions, or using substitution, as illustrated in the step-by-step solution, to simplify each side in such a way that they clearly match, concluding that the identity holds true.

Properties of Trigonometric Functions

Understanding the fundamental properties of trigonometric functions is essential not only in solving equations but also in proving identities. Each function has unique properties that can help in simplifying and manipulating expressions.
Key properties include:

• Periodicity: Functions like sine and cosine repeat their values over intervals, and knowing these periods can assist in transformation or simplification.
• Reciprocal relationships: Functions like sine, cosine, and tangent have reciprocals, which are cosecant, secant, and cotangent respectively. This relation helps in transforming expressions.
• Symmetry: As previously discussed, understanding which functions are even or odd helps in dealing with negative angles.

Using these properties allows us to manipulate trigonometric expressions by simplifying common angles or even working with complex identities by breaking them down into a series of simpler components. This was crucial in our exercise to verify that \( \cos(-t) \csc(-t) = -\cot t \). Each property enables us to transform complex trigonometric identities into forms that are easier to work with, corroborating the balance or truth of a given identity.