Problem 149 - Calculus Volume 1

Question

For the following exercises, verify that each equation is an identity. $$\frac{\sin t}{\csc t}+\frac{\cos t}{\sec t}=1$$

Step-by-Step Solution

Answer

The equation \(\frac{\sin t}{\csc t}+\frac{\cos t}{\sec t}=1\) is an identity.

Step 1: Rewrite Trigonometric Functions

To simplify the equation, we first rewrite the reciprocal trigonometric functions in terms of sine and cosine. The cosecant function, \(\csc t\), is the reciprocal of sine, i.e., \(\csc t = \frac{1}{\sin t}\). Similarly, the secant function, \(\sec t\), is the reciprocal of cosine, i.e., \(\sec t = \frac{1}{\cos t}\).

Step 2: Simplify Each Fraction

Substitute the reciprocal forms into the equation: \[\frac{\sin t}{\csc t} = \sin t \times \sin t = \sin^2 t\]\[\frac{\cos t}{\sec t} = \cos t \times \cos t = \cos^2 t\]This simplifies the original equation to:\[\sin^2 t + \cos^2 t = 1\]

Step 3: Apply Pythagorean Identity

Recall the fundamental Pythagorean identity: \(\sin^2 t + \cos^2 t = 1\). This confirms the equation as an identity. Since both sides of the equation are equal, the original equation is verified as an identity.

Key Concepts

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions are just a different way of looking at the same angle's sine and cosine. They help us see relationships and simplify expressions. The main reciprocal functions are cosecant (\(\csc t\)) and secant (\(\sec t\)), which relate directly to sine and cosine.

• Cosecant is the reciprocal of sine, so \(\csc t = \frac{1}{\sin t}\).
• Secant is the reciprocal of cosine, so \(\sec t = \frac{1}{\cos t}\).

These identities come in handy when breaking down expressions into their most basic forms. Specifically, they allow us to express something complex in terms we are more familiar with. If you ever see \(\csc t\) or \(\sec t\), try writing them as \(\frac{1}{\sin t}\) and \(\frac{1}{\cos t}\), respectively. It can make your problems easier to tackle.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions means transforming a complex expression into a simpler form that is easier to work with. In the given problem, the original equation \(\frac{\sin t}{\csc t} + \frac{\cos t}{\sec t} = 1\) can be simplified by expressing everything in sine and cosine terms. This is achieved by replacing the reciprocal functions with their definitions:

• \(\frac{\sin t}{\csc t} = \sin t \times \sin t = \sin^2 t\)
• \(\frac{\cos t}{\sec t} = \cos t \times \cos t = \cos^2 t\)

By simplifying each fraction independently, you reduce a potentially daunting equation into a simpler expression: \(\sin^2 t + \cos^2 t = 1\). This kind of transformation helps to understand the deeper meaning or properties of the expression and makes solving problems more straightforward.

Pythagorean Identity

The Pythagorean identity is one of the core principles in trigonometry. It relates the square of sine to the square of cosine, forming a direct connection with the Pythagorean Theorem in geometry. The identity is written as:
\[\sin^2 t + \cos^2 t = 1\]
This simple equation expresses an elegant truth about the relationship between the sine and cosine of any angle \(t\). It’s universally true for all angles. In our example, once we have simplified the equation to \(\sin^2 t + \cos^2 t\), recognizing that this equals 1, we can confidently confirm that the original equation is indeed an identity.

• This identity simplifies many trigonometric problems and reveals underlying patterns.
• It’s often used to prove other identities or during integration and differentiation of trigonometric functions.

Memorizing this identity is crucial because it applies in so many scenarios and often serves as a stepping stone in more complicated proofs and solutions.