Problem 13 - Precalculus: Functions and Graphs

Question

Verify the Identity. $$\csc ^{4} t-\cot ^{4} t=\csc ^{2} t+\cot ^{2} t$$

Step-by-Step Solution

Answer

The identity is verified: \( \csc^{4} t - \cot^{4} t = \csc^{2} t + \cot^{2} t \).

Step 1: Understanding the Identity

We are given the trigonometric identity \( \csc^{4} t - \cot^{4} t = \csc^{2} t + \cot^{2} t \). Our goal is to verify whether this is true. To do this, we'll rewrite the left-hand side and simplify it to see if it matches the right-hand side.

Step 2: Rewrite the Left-Hand Side

To simplify \( \csc^{4} t - \cot^{4} t \), use the difference of squares: \( a^2 - b^2 = (a-b)(a+b) \). Here, set \( a = \csc^{2} t \) and \( b = \cot^{2} t \), so:\[ \csc^{4} t - \cot^{4} t = (\csc^{2} t - \cot^{2} t)(\csc^{2} t + \cot^{2} t) \]

Step 3: Evaluate \( \csc^{2} t - \cot^{2} t \)

We know from trigonometric identities that \( \csc^{2} t = 1 + \cot^{2} t \). Substitute this into the expression:\[ \csc^{2} t - \cot^{2} t = (1 + \cot^{2} t) - \cot^{2} t = 1 \]

Step 4: Simplify the Expression

Substitute \( \csc^{2} t - \cot^{2} t = 1 \) back into the identity from Step 2:\[ (1)(\csc^{2} t + \cot^{2} t) = \csc^{2} t + \cot^{2} t \]

Step 5: Verify the Equality

The simplified left-hand side \( \csc^{2} t + \cot^{2} t \) matches exactly the right-hand side \( \csc^{2} t + \cot^{2} t \). Thus, the original identity is verified to be true.

Key Concepts

Cosecant and Cotangent Identities

Cosecant and cotangent are two important trigonometric functions. They are related to the more common sine and cosine functions. Understanding their identities is crucial for solving various trigonometric problems. Let's start with the basic definitions:

• Cosecant (\( \csc t \)) is the reciprocal of the sine function, so \( \csc t = \frac{1}{\sin t} \).
• Cotangent (\( \cot t \)) is the reciprocal of the tangent function and can also be expressed as the ratio of \( \cos t \) to \( \sin t \), so \( \cot t = \frac{\cos t}{\sin t} \).

Trigonometric identities provide relationships between these functions. An important identity involving these functions is:\[\csc^2 t = 1 + \cot^2 t\]This identity allows us to express the cosecant squared function in terms of cotangent squared and is often used in simplifying expressions like the one in this problem. Understanding how to use these identities to convert expressions is key to mastering trigonometric equations.

Difference of Squares

The difference of squares is a powerful algebraic tool that helps in simplifying expressions. It states that the difference between two squares can be expressed as a product of the difference and sum of their bases.When dealing with an expression of the form \( a^2 - b^2 \), this can be rewritten as:\[(a-b)(a+b)\]In our problem, we applied this by setting \( a = \csc^2 t \) and \( b = \cot^2 t \). Consequently, the expression \( \csc^4 t - \cot^4 t \) becomes:\[(\csc^2 t - \cot^2 t)(\csc^2 t + \cot^2 t)\]This step is necessary because it reduces the expression into a product. Once simplified, further substitutions using trigonometric identities can take place. Recognizing patterns like this difference of squares can greatly facilitate problem-solving, reducing complexity and revealing simpler forms.

Simplification of Trigonometric Expressions

Simplifying trigonometric expressions is crucial in verifying identities and solving equations. The ultimate goal is to transform an expression into its simplest form, often using trigonometric identities and algebraic techniques.In our example, we started with:\[\csc^4 t - \cot^4 t\]After rewriting using the difference of squares, we simplified further using the identity \( \csc^2 t = 1 + \cot^2 t \). By substituting \( \csc^2 t \) into the expression:\[\csc^2 t - \cot^2 t = 1\]This simplification led us directly to verifying the identity, because it enabled the cancellation of terms. Therefore, the expression became clearly identical on both sides. Simplification involves not only recognizing algebraic patterns but also knowing when and how to apply trigonometric identities, which are essential strategies in trigonometry.