Problem 8 - Precalculus with Calculus Previews

Question

Use the fundamental identities and the even-odd identities to simplify each expression. $$ \cos ^{2} t+\frac{1}{\csc ^{2} t} $$

Step-by-Step Solution

Answer

The expression simplifies to 1.

Step 1: Rewrite the Expression with Fundamental Identities

The original expression is \( \cos^{2} t + \frac{1}{\csc^{2} t} \).First, recall the identity for cosecant: \( \csc t = \frac{1}{\sin t} \). This means \( \csc^{2} t = \frac{1}{\sin^{2} t} \).Thus, \( \frac{1}{\csc^{2} t} \) can be rewritten as \( \sin^{2} t \).Now the expression becomes: \( \cos^{2} t + \sin^{2} t \).

Step 2: Apply the Pythagorean Identity

According to the Pythagorean Identity, \( \sin^{2} t + \cos^{2} t = 1 \).Thus, the expression \( \cos^{2} t + \sin^{2} t \) simplifies to 1.

Key Concepts

Fundamental Identities

Trigonometric identities provide the foundation for simplifying and understanding trigonometric expressions. Among these, the fundamental identities are of great importance. Let's delve into them. The fundamental identities include basic relationships between trigonometric functions. These are crucial for transforming and manipulating expressions. A vital identity in this group is:

• The reciprocal identities, such as \(\csc t = \frac{1}{\sin t}\), which indicates that cosecant is the reciprocal of the sine function.
• The identity allows us to rewrite expressions in different forms, which can greatly simplify complex problems.
• By using these identities, you can transform \(\frac{1}{\csc^2 t}\) into \(\sin^2 t\).

Fundamental identities serve as building blocks in trigonometry. Recognizing and applying them effectively is key to mastering this mathematical area.

Even-Odd Identities

Even-odd identities in trigonometry tell us about the symmetry of trigonometric functions. They help simplify expressions and solve equations involving sines and cosines. Here's how:

• Even functions, like \(\cos(x)\), maintain the same value when the input angle changes sign, i.e., \(\cos(-x) = \cos(x)\).
• Odd functions, like \(\sin(x)\), change sign when the input angle changes sign, i.e., \(\sin(-x) = -\sin(x)\).
• Even-odd identities allow easy transformation of functions with negative angles or inputs.

Recognizing whether a function is even or odd can simplify the integration process and aid in understanding the behavior of functions across different quadrants.

Pythagorean Identity

The Pythagorean identity is arguably one of the most useful identities in trigonometry. It comes directly from the Pythagorean theorem applied to circles on the unit circle. The identity is: \[ \sin^2 t + \cos^2 t = 1 \] This lets us express one trigonometric function in terms of another. Let's explore its significance:

• It's derived from the equation of a circle, \(x^2 + y^2 = 1\), with \(x = \cos t \) and \(y = \sin t\).
• The identity is pivotal for converting between different trigonometric forms.
• It confirms that whenever you have a squared sine and cosine of the same angle, their sum is always 1.

In the original exercise, the expression simplifies straightforwardly using this identity. Recognizing this can help in simplifying expressions rapidly without cumbersome calculations.