Problem 31 - Algebra and Trigonometry with Analytic Geometry

Question

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

Step-by-Step Solution

Answer

The identity is verified as (\(\csc t - \cot t\))^4(\(\csc t + \cot t\))^4 equals 1.

Step 1: Understand the Identity to Verify

We have the identity equation \((\csc t - \cot t)^4(\csc t + \cot t)^4 = 1\). Our task is to simplify the left-hand side and demonstrate that it equals 1.

Step 2: Substitute Trigonometric Identities

Recall that \(\csc t = \frac{1}{\sin t}\) and \(\cot t = \frac{\cos t}{\sin t}\). Substitute these into the expression:\[\left(\frac{1}{\sin t} - \frac{\cos t}{\sin t}\right)^4 \left(\frac{1}{\sin t} + \frac{\cos t}{\sin t}\right)^4.\]

Step 3: Simplify the Expression Inside Parentheses

Factor out \(\frac{1}{\sin t}\) in both terms inside the parentheses:\[\left(\frac{1 - \cos t}{\sin t}\right)^4 \left(\frac{1 + \cos t}{\sin t}\right)^4.\]

Step 4: Use the Pythagorean Identity

We know the Pythagorean identity \(1 - \cos^2 t = \sin^2 t\). Notice that \((1-\cos t)(1+\cos t) = 1 - \cos^2 t = \sin^2 t\). Thus, \[(1 - \cos t)^4 (1 + \cos t)^4 = (\sin^2 t)^4.\]

Step 5: Combine the Factors

Now, the expression becomes:\[\left( \frac{(\sin^2 t)^4}{\sin^8 t} \right) = 1.\]Since \(\sin^8 t\) cancels the numerator's \(\sin^8 t\), this results in 1.

Key Concepts

Cosecant (csc)

Cosecant, often abbreviated as \( \csc \), is one of the six fundamental trigonometric functions. It is defined as the reciprocal of the sine function. This means that for an angle \( t \), the cosecant is calculated as:

• \( \csc t = \frac{1}{\sin t} \)

This function is especially useful when dealing with trigonometric identities as it often appears in expressions that involve division by sine. The cosecant has certain key properties:

• It is undefined when \( \sin t = 0 \), which occurs at integer multiples of \( \pi \), because you cannot divide by zero.
• Cosecant values are always more than 1 or less than -1 because sine values range between -1 and 1, and their reciprocals thus fall outside this range.

Understanding the function's behavior helps when substituting it into trigonometric identities during simplifications or verifications.

Cotangent (cot)

The cotangent, abbreviated as \( \cot \), is another important trigonometric function. It is defined as the reciprocal of the tangent function or the ratio of cosine to sine. Specifically, for an angle \( t \), it is given by:

• \( \cot t = \frac{1}{\tan t} = \frac{\cos t}{\sin t} \)

The cotangent function is crucial for expressing trigonometric identities and can simplify complex trigonometric expressions. Here are some notable characteristics:

• It is undefined whenever \( \sin t = 0 \), corresponding to angles like \( 0, \pi, 2\pi \), and so on, similar to the cosecant function.
• It can take any real number as its value depending on the angle \( t \), since neither \( \cos t \) nor \( \sin t \) is bound between -1 and 1 when \( \sin t eq 0 \).

Cotangent is particularly useful in problems involving right triangles and in converting complex trigonometric expressions into simpler or more manageable forms.

Pythagorean Identity

The Pythagorean identity is a fundamental relation in trigonometry that connects the sine and cosine functions. It states:

• \( \sin^2 t + \cos^2 t = 1 \)

This identity forms the basis for deriving other important trigonometric identities, such as:

• \( 1 - \cos^2 t = \sin^2 t \)
• \( 1 - \sin^2 t = \cos^2 t \)

In our given exercise, the Pythagorean identity helps in simplifying the expression: \( (1 - \cos t)(1 + \cos t) = \sin^2 t \). This fact is then utilized to achieve the result \( 1 \) by resting on its power-depend relation. Recognizing variations and how to apply or rearrange this identity is critical for solving many trigonometric problems.