Problem 41 - Algebra and Trigonometry

Question

Verify the identity. $$ \frac{\sec t-\cos t}{\sec t}=\sin ^{2} t $$

Step-by-Step Solution

Answer

The identity is verified: \( \frac{\sec t - \cos t}{\sec t} = \sin^2 t \).

Step 1: Understand the Identity

The exercise asks us to verify the trigonometric identity: \( \frac{\sec t - \cos t}{\sec t} = \sin^2 t \). To verify, we'll manipulate the left-hand side (LHS) to simplify it into the right-hand side (RHS).

Step 2: Convert to Basic Trigonometric Functions

Recall that \( \sec t = \frac{1}{\cos t} \). Substituting \( \sec t \) in the identity, the LHS becomes: \( \frac{\frac{1}{\cos t} - \cos t}{\frac{1}{\cos t}} \).

Step 3: Simplify the Numerator

Subtract the terms in the numerator: \( \frac{1}{\cos t} - \cos t = \frac{1 - \cos^2 t}{\cos t} \). Recognizing that \( 1 - \cos^2 t = \sin^2 t \) from the Pythagorean identity.

Step 4: Combine the Terms

So, the LHS is \( \frac{\sin^2 t}{\cos t} \) divided by \( \frac{1}{\cos t} \). Simplifying this gives: \( \sin^2 t \cdot \cos t \cdot \frac{\cos t}{1} = \sin^2 t \).

Step 5: Match the Right-Hand Side

After simplification, the LHS becomes \( \sin^2 t \), which is identical to the RHS. Hence, the identity is verified.

Key Concepts

Pythagorean Identity

The Pythagorean identity is one of the most fundamental trigonometric identities and forms the basis of many verification exercises, like the one given here. It states that for any angle \( t \), the relation \( \sin^2 t + \cos^2 t = 1 \) holds true. This identity is derived from the Pythagorean theorem applied to the unit circle, where the radius is 1. When calculating these values on a unit circle:

• \( \cos t \) and \( \sin t \) represent the horizontal and vertical distances, respectively.
• The radius of the circle (hypotenuse) remains 1.

When you rearrange this identity, you get other useful expressions like \( 1 - \cos^2 t = \sin^2 t \), which was used in simplifying the original exercise. Recognizing and applying the Pythagorean identity will make solving similar trigonometric problems much easier.

Secant and Cosine Relationship

The secant and cosine relationship is essential in simplifying trigonometric expressions. Understanding this relationship is crucial as it helps in conversions and substitutions in trigonometric equations. The secant function is defined as the reciprocal of the cosine function:

• \( \sec t = \frac{1}{\cos t} \)

This means that wherever you see \( \sec t \), you can substitute it with \( \frac{1}{\cos t} \). In the given exercise, recognizing this relationship allowed us to convert the expression into basic cosine and sine terms. It facilitated the simplification process by transforming both functions into terms having a common cosine denominator, making it easier to apply the Pythagorean identity.

Verification of Identities

Verification of trigonometric identities involves proving that two sides of an equation are equivalent by transformation or simplification. This process is like solving a puzzle where you reshape and reformat the pieces until they match.
In the problem we tackled, verifying was done by manipulating the left-hand side of the equation to resemble the right-hand side:

• Start with rewriting or substituting functions for simplification.
• Look for opportunities to apply known identities, like the Pythagorean identity.
• Simplify each step carefully to match both sides of the equation.

This method not only builds your confidence in solving such problems but also deepens your understanding of how these identities interconnect and operate within the realm of trigonometry.