Problem 13 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-50: Verify the identity. $$ \csc ^{4} t-\cot ^{4} t=\csc ^{2} t+\cot ^{2} t $$

Step-by-Step Solution

Answer

The identity is verified. The equation is true.

Step 1: Understanding the Identity

We want to verify if the equation \( ext{csc}^4 t - ext{cot}^4 t = ext{csc}^2 t + ext{cot}^2 t\) holds true. We'll start by transforming one side of the equation to match the other side.

Step 2: Express in Terms of Sine and Cosine

We know that \( ext{csc}^2 t = \frac{1}{\sin^2 t}\) and \( ext{cot}^2 t = \frac{\cos^2 t}{\sin^2 t}\). We'll use these relationships to re-express \( ext{csc}^4 t - ext{cot}^4 t\) in terms of sine and cosine.

Step 3: Simplify Left Side

Start by factoring \( ext{csc}^4 t - ext{cot}^4 t\) as a difference of squares: \[( ext{csc}^2 t - ext{cot}^2 t)( ext{csc}^2 t + ext{cot}^2 t)\]. Since we need to match the identity, we will focus on simplifying \( ext{csc}^2 t - ext{cot}^2 t\).

Step 4: Simplify \( ext{csc}^2 t - ext{cot}^2 t\)

Substitute the identity \( ext{csc}^2 t = 1 + ext{cot}^2 t\): \[ ext{csc}^2 t - ext{cot}^2 t = (1 + ext{cot}^2 t) - ext{cot}^2 t = 1\].

Step 5: Complete Simplification

Now substitute back into the factored form: \[( ext{csc}^2 t - ext{cot}^2 t)( ext{csc}^2 t + ext{cot}^2 t) = 1 imes ( ext{csc}^2 t + ext{cot}^2 t) = ext{csc}^2 t + ext{cot}^2 t\].This shows the original equation holds true.

Key Concepts

Cosecant Function

The cosecant function is one of the six fundamental trigonometric functions. It is the reciprocal of the sine function. This means that \[ \csc t = \frac{1}{\sin t} \]. Since the sine function can often have very small values, the cosecant can balloon to large numbers, especially as it approaches angles where sine equals zero.

• Important for solving trigonometric identities.
• Useful in context with the cotangent function.

Understanding the cosecant is crucial when transforming and simplifying expressions. Cosecant squared, denoted as \( \csc^2 t \), comes up often in identities, such as in the given exercise where it is expressed in relation to cotangent.

Cotangent Function

The cotangent function is closely related to the tangent but is its reciprocal. This means that \( \cot t = \frac{\cos t}{\sin t} \). Like cosecant, cotangent can also take on large values, especially near points where tangent becomes zero, due to division by small sine values.

• Reciprocal of the tangent function.
• Appears frequently in engineering and physics problems.

In trigonometric identities, cotangent often teams with cosecant, notably in expressions like \( 1 + \cot^2 t = \csc^2 t \). These relationships help in verifying identities by transforming expressions into simpler or known forms.

Trigonometric Simplification

Trigonometric simplification involves using known identities to transform expressions into equivalent forms that might be easier to work with. It is crucial for solving equations or verifying identities.

• Relies on algebraic manipulation and trigonometric identities.
• Methods include factoring, using reciprocal identities, and leveraging known formulas like \( \sin^2 t + \cos^2 t = 1 \)

In the given exercise, simplification is seen through the identity \( \csc^2 t - \cot^2 t = 1 \). By recognizing and applying these simple yet profound identities, complex expressions can be simplified, making them easier to understand and verify.