To Verify The Identity, Start With The More Complicated Side And Transform It To Look Like The Other Side. Choose The Correct Transformations And Transform The Expression At Each Step.$\[ \begin{aligned} \frac{\cot^2 T}{\csc T} & =

Introduction

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is a fundamental subject that has numerous applications in various fields, including physics, engineering, and navigation. In this article, we will focus on simplifying trigonometric expressions, which is an essential skill for any student of mathematics.

Understanding the Problem

The given problem is to simplify the expression $\frac{\cot^2 t}{\csc t}$. To do this, we need to apply various trigonometric identities and transformations to transform the expression into a simpler form.

Step 1: Simplify the Cotangent Term

The cotangent function is defined as $\cot t = \frac{\cos t}{\sin t}$. We can rewrite the cotangent term in the given expression as follows:

$$\cot^2 t = \left(\frac{\cos t}{\sin t}\right)^2 = \frac{\cos^2 t}{\sin^2 t}$$

Step 2: Simplify the Cosecant Term

The cosecant function is defined as $\csc t = \frac{1}{\sin t}$. We can rewrite the cosecant term in the given expression as follows:

$$\csc t = \frac{1}{\sin t}$$

Step 3: Substitute the Simplified Terms

Now that we have simplified the cotangent and cosecant terms, we can substitute them into the original expression:

$$\frac{\cot^2 t}{\csc t} = \frac{\frac{\cos^2 t}{\sin^2 t}}{\frac{1}{\sin t}}$$

Step 4: Simplify the Expression

To simplify the expression, we can multiply the numerator and denominator by $\sin t$:

$$\frac{\frac{\cos^2 t}{\sin^2 t}}{\frac{1}{\sin t}} = \frac{\cos^2 t}{\sin^2 t} \cdot \sin t = \frac{\cos^2 t}{\sin t}$$

Step 5: Apply the Pythagorean Identity

The Pythagorean identity states that $\sin^2 t + \cos^2 t = 1$. We can rewrite the expression in terms of this identity:

$$\frac{\cos^2 t}{\sin t} = \frac{1 - \sin^2 t}{\sin t}$$

Step 6: Simplify the Expression

To simplify the expression, we can multiply the numerator and denominator by $\sin t$:

$$\frac{1 - \sin^2 t}{\sin t} = \frac{\sin t - \sin^3 t}{\sin^2 t}$$

Step 7: Factor Out the Common Term

We can factor out the common term $\sin t$ from the numerator:

$$\frac{\sin t - \sin^3 t}{\sin^2 t} = \frac{\sin t(1 - \sin^2 t)}{\sin^2 t}$$

Step 8: Cancel Out the Common Term

We can cancel out the common term $\sin t$ from the numerator and denominator:

$$\frac{\sin t(1 - \sin^2 t)}{\sin^2 t} = \frac{1 - \sin^2 t}{\sin t}$$

Step 9: Apply the Pythagorean Identity Again

We can apply the Pythagorean identity again to simplify the expression:

$$\frac{1 - \sin^2 t}{\sin t} = \frac{\cos^2 t}{\sin t}$$

Step 10: Simplify the Expression

To simplify the expression, we can multiply the numerator and denominator by $\sin t$:

$$\frac{\cos^2 t}{\sin t} = \frac{\cos^2 t \sin t}{\sin^2 t}$$

Step 11: Cancel Out the Common Term

We can cancel out the common term $\sin t$ from the numerator and denominator:

$$\frac{\cos^2 t \sin t}{\sin^2 t} = \frac{\cos^2 t}{\sin t}$$

Conclusion

In this article, we have simplified the expression $\frac{\cot^2 t}{\csc t}$ using various trigonometric identities and transformations. We have applied the Pythagorean identity, factored out common terms, and canceled out common terms to simplify the expression. The final simplified expression is $\frac{\cos^2 t}{\sin t}$.

Final Answer

$$

Q&A: Simplifying Trigonometric Expressions

Q: What is the first step in simplifying a trigonometric expression? A: The first step in simplifying a trigonometric expression is to identify the trigonometric functions involved and to rewrite them in terms of sine and cosine.

Q: How do I simplify a cotangent term? A: To simplify a cotangent term, you can rewrite it as $\cot t = \frac{\cos t}{\sin t}$ and then simplify the resulting expression.

Q: How do I simplify a cosecant term? A: To simplify a cosecant term, you can rewrite it as $\csc t = \frac{1}{\sin t}$ and then simplify the resulting expression.

Q: What is the Pythagorean identity? A: The Pythagorean identity states that $\sin^2 t + \cos^2 t = 1$. This identity can be used to simplify trigonometric expressions.

Q: How do I apply the Pythagorean identity to simplify an expression? A: To apply the Pythagorean identity, you can substitute $\sin^2 t + \cos^2 t = 1$ into the expression and then simplify the resulting expression.

Q: What is the difference between factoring and canceling out common terms? A: Factoring involves expressing an expression as a product of simpler expressions, while canceling out common terms involves eliminating common factors from the numerator and denominator of an expression.

Q: How do I factor out a common term? A: To factor out a common term, you can identify the common term and then express the expression as a product of the common term and a simpler expression.

Q: How do I cancel out common terms? A: To cancel out common terms, you can identify the common terms in the numerator and denominator of an expression and then eliminate them.

Q: What are some common trigonometric identities that I should know? A: Some common trigonometric identities that you should know include the Pythagorean identity, the sum and difference formulas, and the double-angle and half-angle formulas.

Q: How do I use trigonometric identities to simplify expressions? A: To use trigonometric identities to simplify expressions, you can substitute the identity into the expression and then simplify the resulting expression.

Q: What are some tips for simplifying trigonometric expressions? A: Some tips for simplifying trigonometric expressions include:

• Identifying the trigonometric functions involved and rewriting them in terms of sine and cosine
• Applying the Pythagorean identity and other trigonometric identities to simplify the expression
• Factoring out common terms and canceling out common terms
• Using trigonometric identities to simplify the expression

Conclusion

In this article, we have provided a step-by-step guide to simplifying trigonometric expressions, including a Q&A section that answers common questions about simplifying trigonometric expressions. We have also provided tips for simplifying trigonometric expressions and have discussed common trigonometric identities that you should know.

Final Answer

The final answer is $\boxed{\frac{\cos^2 t}{\sin t}}$.