Problem 59 - Algebra and Trigonometry

Question

Verify the identity. $$ \frac{(\sin t+\cos t)^{2}}{\sin t \cos t}=2+\sec t \csc t $$

Step-by-Step Solution

Answer

The identity is verified: both sides simplify to the same expression.

Step 1: Expand the Numerator

First, expand the numerator \((\sin t + \cos t)^2\). This can be done using the formula for the square of a binomial: \((a + b)^2 = a^2 + 2ab + b^2\). Substitute \(a = \sin t\) and \(b = \cos t\): \[(\sin t + \cos t)^2 = \sin^2 t + 2\sin t \cos t + \cos^2 t.\] Since \(\sin^2 t + \cos^2 t = 1\), this simplifies to: \[1 + 2\sin t \cos t.\]

Step 2: Rewrite the Expression

Substitute back the expanded form into the original fraction: \[\frac{1 + 2\sin t \cos t}{\sin t \cos t}.\] Split the fraction into two separate terms: \[\frac{1}{\sin t \cos t} + \frac{2\sin t \cos t}{\sin t \cos t}.\] This simplifies to: \[\frac{1}{\sin t \cos t} + 2.\]

Step 3: Simplify the Fraction

Observe that \(\frac{1}{\sin t \cos t}\) can be rewritten using the identities for secant and cosecant: \[\frac{1}{\sin t \cos t} = \frac{1}{\sin t} \cdot \frac{1}{\cos t} = \csc t \cdot \sec t = \sec t \csc t.\]

Step 4: Combine the Terms

Using the simplification from Step 3, the expression from Step 2 is now: \[2 + \sec t \csc t.\]

Step 5: Verify the Identity

Compare the original identity \(\frac{(\sin t+\cos t)^{2}}{\sin t \cos t}=2+\sec t \csc t\) with the simplified expression from Step 4. They match, confirming that the identity is verified.

Key Concepts

Binomial Expansion

The binomial expansion is a crucial algebraic technique used to expand expressions that are raised to a power. It allows us to break down expressions like

• \((a + b)^n\)

into manageable terms. For example, when we expand

• \(( ext{sin} t + ext{cos} t)^2\),

we apply the formula

• \((a + b)^2 = a^2 + 2ab + b^2\),

which results in the expression

• \( ext{sin}^2 t + 2 ext{sin} t ext{cos} t + ext{cos}^2 t\).

This formula helps to simplify further trigonometric identities as we utilize the fundamental identity:

• \( ext{sin}^2 t + ext{cos}^2 t = 1\),

leading to a simplified form of the expanded expression. This step reveals the power of binomial expansion in solving complex mathematical expressions, especially those involving trigonometric functions.

Secant Function

The secant function is one of the six fundamental trigonometric functions. It is defined as the reciprocal of the cosine function. In mathematical terms:

• \( ext{sec} t = \frac{1}{ ext{cos} t}\).

The secant function is particularly useful in various trigonometric simplifications and transformations. When evaluating expressions, the secant function often appears in identities and equations that can be simplified by expressing them in terms of other trigonometric functions. For instance, in the given exercise,

• \( ext{sec} t ext{ } ext{csc} t\)

is used as a simplified form of a trigonometric identity, offering a compact and simplified version of the problem to be worked on.

Cosecant Function

The cosecant function is another key trigonometric function, defined as the reciprocal of the sine function. Its mathematical representation is:

• \( ext{csc} t = \frac{1}{ ext{sin} t}\).

Like the secant function, the cosecant function plays a significant role in simplifying trigonometric expressions and identities. It often appears in problems that involve division by sine, allowing for the simplification of expressions into more manageable forms. Specifically, in our exercise, the cosecant function was crucial in transforming the complex fraction into a simpler equivalent using trigonometric identities to verify the original expression. It demonstrates the importance of recognizing how the cosecant ties into broader trigonometric relationships.

Trigonometric Simplification

Trigonometric simplification involves using identities and algebraic manipulation to reduce complex trigonometric expressions into simpler ones. This process is crucial for proving identities, solving equations, and verifying trigonometric statements.
In our step-by-step solution, we see trigonometric simplification in action. We start by expanding the numerator using the binomial expansion, followed by breaking up the expression into simpler fractions:

• \(\frac{1 + 2 ext{sin} t ext{cos} t}{ ext{sin} t ext{cos} t} = \frac{1}{ ext{sin} t ext{cos} t} + 2\).

By recognizing relationships such as

• \(\frac{1}{ ext{sin} t ext{cos} t} = ext{sec} t ext{csc} t\),

we use these identities to simplify the fraction, effectively confirming the original identity statement. This demonstrates how systematic simplification can unravel even the most complex-looking trigonometric expressions, turning them into recognizable and verifiable identities.