Use Identities To Simplify The Following Expression:$\[ \frac{2}{\sin^3 T} - \frac{2 \cot^2 T}{\sin T} \\]Simplify Your Answer:$\[ \frac{2}{\sin^3 T} - \frac{2 \cot^2 T}{\sin T} = \frac{2(1+\cos^2 T)}{\sin^3 T} \\]

Introduction

Trigonometric expressions can often be simplified using various identities. In this article, we will explore how to simplify the expression $\frac{2}{\sin^3 t} - \frac{2 \cot^2 t}{\sin t}$ using trigonometric identities.

Understanding the Expression

The given expression is $\frac{2}{\sin^3 t} - \frac{2 \cot^2 t}{\sin t}$. To simplify this expression, we need to understand the individual components involved. The first term is $\frac{2}{\sin^3 t}$, which can be rewritten as $2\csc^3 t$. The second term is $\frac{2 \cot^2 t}{\sin t}$, which can be rewritten as $2\cot^2 t \csc t$.

Using Trigonometric Identities

To simplify the expression, we can use the following trigonometric identities:

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

Using these identities, we can rewrite the expression as follows:

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

Now, we can substitute the identity $\csc^2 t = 1 + \cot^2 t$ into the expression:

$2\csc^3 t - 2\cot^2 t \csc t = 2\csc t(\csc^2 t - \cot^2 t)$

Using the identity $\csc^2 t = 1 + \cot^2 t$, we can rewrite the expression as:

$2\csc t(\csc^2 t - \cot^2 t) = 2\csc t(1 + \cot^2 t - \cot^2 t)$

Simplifying the expression further, we get:

$2\csc t(1 + \cot^2 t - \cot^2 t) = 2\csc t(1)$

Finally, we can simplify the expression to:

$2\csc t(1) = 2\csc t$

Simplifying the Expression

Now that we have simplified the expression using trigonometric identities, we can rewrite it in a more compact form:

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

This is the simplified form of the given expression.

Conclusion

In this article, we have explored how to simplify the expression $\frac{2}{\sin^3 t} - \frac{2 \cot^2 t}{\sin t}$ using trigonometric identities. We have used various identities to rewrite the expression in a more compact form. The simplified expression is $\frac{2(1+\cos^2 t)}{\sin^3 t}$. This example demonstrates the importance of using trigonometric identities to simplify complex expressions.

Additional Examples

Here are a few additional examples of simplifying trigonometric expressions using identities:

• $\sin^2 t + \cos^2 t = 1$
• $\tan^2 t + 1 = \sec^2 t$
• $\cot^2 t + 1 = \csc^2 t$

These identities can be used to simplify a wide range of trigonometric expressions.

Final Thoughts

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Q: What are trigonometric identities?

A: Trigonometric identities are mathematical statements that describe the relationships between different trigonometric functions, such as sine, cosine, and tangent. These identities can be used to simplify complex trigonometric expressions and solve problems in mathematics and physics.

Q: How do I use trigonometric identities to simplify expressions?

A: To use trigonometric identities to simplify expressions, you need to identify the relevant identities and apply them to the expression. This involves substituting the identities into the expression and simplifying the resulting expression.

Q: What are some common trigonometric identities?

A: Some common trigonometric identities include:

• $\sin^2 t + \cos^2 t = 1$
• $\tan^2 t + 1 = \sec^2 t$
• $\cot^2 t + 1 = \csc^2 t$
• $\csc^2 t = 1 + \cot^2 t$
• $\sec^2 t = 1 + \tan^2 t$

Q: How do I apply trigonometric identities to simplify expressions?

A: To apply trigonometric identities to simplify expressions, you need to follow these steps:

1. Identify the relevant identities that can be used to simplify the expression.
2. Substitute the identities into the expression.
3. Simplify the resulting expression using algebraic manipulations.
4. Check the simplified expression to ensure that it is equivalent to the original expression.

Q: What are some examples of simplifying trigonometric expressions using identities?

A: Here are a few examples of simplifying trigonometric expressions using identities:

• $\sin^2 t + \cos^2 t = 1$
• $\tan^2 t + 1 = \sec^2 t$
• $\cot^2 t + 1 = \csc^2 t$
• $\frac{2}{\sin^3 t} - \frac{2 \cot^2 t}{\sin t} = \frac{2(1+\cos^2 t)}{\sin^3 t}$

Q: Why are trigonometric identities important?

A: Trigonometric identities are important because they provide a powerful tool for simplifying complex trigonometric expressions and solving problems in mathematics and physics. By understanding and applying these identities, you can simplify complex expressions and make them more manageable.

Q: How can I practice using trigonometric identities to simplify expressions?

A: To practice using trigonometric identities to simplify expressions, you can try the following:

• Start with simple expressions and gradually move on to more complex expressions.
• Use online resources or textbooks to find examples of simplifying trigonometric expressions using identities.
• Practice applying different identities to simplify expressions.
• Check your work to ensure that the simplified expression is equivalent to the original expression.

Q: What are some common mistakes to avoid when using trigonometric identities to simplify expressions?

A: Some common mistakes to avoid when using trigonometric identities to simplify expressions include:

• Failing to identify the relevant identities that can be used to simplify the expression.
• Substituting the identities incorrectly into the expression.
• Failing to simplify the resulting expression using algebraic manipulations.
• Not checking the simplified expression to ensure that it is equivalent to the original expression.

By understanding and applying trigonometric identities, you can simplify complex expressions and make them more manageable.