Simplify The Expression:${ \csc^2 \theta \tan^2 \theta - 1 = \tan^2 \theta }$

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Introduction

In trigonometry, simplifying expressions involving trigonometric functions is a crucial skill. The given expression, csc⁡2θtan⁡2θ−1=tan⁡2θ\csc^2 \theta \tan^2 \theta - 1 = \tan^2 \theta, appears to be a complex equation involving the cosecant and tangent functions. In this article, we will simplify this expression step by step, using various trigonometric identities and formulas.

Understanding the Trigonometric Functions

Before we dive into simplifying the expression, let's briefly review the trigonometric functions involved:

  • Cosecant (csc): The cosecant function is the reciprocal of the sine function. It is defined as csc⁡θ=1sin⁡θ\csc \theta = \frac{1}{\sin \theta}.
  • Tangent (tan): The tangent function is the ratio of the sine and cosine functions. It is defined as tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}.

Simplifying the Expression

To simplify the expression, we can start by using the Pythagorean identity, which states that sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1. We can rewrite this identity as cos⁡2θ=1−sin⁡2θ\cos^2 \theta = 1 - \sin^2 \theta.

# Pythagorean Identity
## cos^2 θ = 1 - sin^2 θ

Step 1: Simplify the Cosecant Function

Using the definition of the cosecant function, we can rewrite csc⁡2θ\csc^2 \theta as 1sin⁡2θ\frac{1}{\sin^2 \theta}.

# Simplify the Cosecant Function
## csc^2 θ = 1 / sin^2 θ

Step 2: Simplify the Tangent Function

Using the definition of the tangent function, we can rewrite tan⁡2θ\tan^2 \theta as sin⁡2θcos⁡2θ\frac{\sin^2 \theta}{\cos^2 \theta}.

# Simplify the Tangent Function
## tan^2 θ = sin^2 θ / cos^2 θ

Step 3: Substitute the Simplified Functions

Now, we can substitute the simplified functions into the original expression:

1sin⁡2θ⋅sin⁡2θcos⁡2θ−1=sin⁡2θcos⁡2θ\frac{1}{\sin^2 \theta} \cdot \frac{\sin^2 \theta}{\cos^2 \theta} - 1 = \frac{\sin^2 \theta}{\cos^2 \theta}

# Substitute the Simplified Functions
## (1 / sin^2 θ) * (sin^2 θ / cos^2 θ) - 1 = sin^2 θ / cos^2 θ

Step 4: Simplify the Expression

We can simplify the expression by canceling out the common factors:

1cos⁡2θ−1=sin⁡2θcos⁡2θ\frac{1}{\cos^2 \theta} - 1 = \frac{\sin^2 \theta}{\cos^2 \theta}

# Simplify the Expression
## 1 / cos^2 θ - 1 = sin^2 θ / cos^2 θ

Step 5: Use the Pythagorean Identity

We can use the Pythagorean identity to rewrite the expression:

1cos⁡2θ−1=1−cos⁡2θcos⁡2θ\frac{1}{\cos^2 \theta} - 1 = \frac{1 - \cos^2 \theta}{\cos^2 \theta}

# Use the Pythagorean Identity
## 1 / cos^2 θ - 1 = (1 - cos^2 θ) / cos^2 θ

Step 6: Simplify the Expression

We can simplify the expression by canceling out the common factors:

1−cos⁡2θcos⁡2θ=sin⁡2θcos⁡2θ\frac{1 - \cos^2 \theta}{\cos^2 \theta} = \frac{\sin^2 \theta}{\cos^2 \theta}

# Simplify the Expression
## (1 - cos^2 θ) / cos^2 θ = sin^2 θ / cos^2 θ

Conclusion

In this article, we simplified the expression csc⁡2θtan⁡2θ−1=tan⁡2θ\csc^2 \theta \tan^2 \theta - 1 = \tan^2 \theta step by step, using various trigonometric identities and formulas. We started by simplifying the cosecant and tangent functions, then substituted the simplified functions into the original expression. Finally, we used the Pythagorean identity to rewrite the expression and simplify it further. The simplified expression is sin⁡2θcos⁡2θ\frac{\sin^2 \theta}{\cos^2 \theta}.

Final Answer

The final answer is sin⁡2θcos⁡2θ\boxed{\frac{\sin^2 \theta}{\cos^2 \theta}}.

References

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Introduction

In our previous article, we simplified the expression csc⁡2θtan⁡2θ−1=tan⁡2θ\csc^2 \theta \tan^2 \theta - 1 = \tan^2 \theta step by step, using various trigonometric identities and formulas. In this article, we will answer some frequently asked questions (FAQs) related to the simplification of this expression.

Q1: What is the Pythagorean identity?

A1: The Pythagorean identity is a fundamental concept in trigonometry that states sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1. This identity can be used to rewrite the expression csc⁡2θtan⁡2θ−1=tan⁡2θ\csc^2 \theta \tan^2 \theta - 1 = \tan^2 \theta in a simpler form.

Q2: How do I simplify the cosecant function?

A2: The cosecant function can be simplified by rewriting it as 1sin⁡2θ\frac{1}{\sin^2 \theta}. This can be done by using the definition of the cosecant function, which is csc⁡θ=1sin⁡θ\csc \theta = \frac{1}{\sin \theta}.

Q3: How do I simplify the tangent function?

A3: The tangent function can be simplified by rewriting it as sin⁡2θcos⁡2θ\frac{\sin^2 \theta}{\cos^2 \theta}. This can be done by using the definition of the tangent function, which is tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}.

Q4: What is the final simplified expression?

A4: The final simplified expression is sin⁡2θcos⁡2θ\frac{\sin^2 \theta}{\cos^2 \theta}.

Q5: Can I use the Pythagorean identity to simplify the expression further?

A5: Yes, you can use the Pythagorean identity to simplify the expression further. By rewriting the expression as 1−cos⁡2θcos⁡2θ\frac{1 - \cos^2 \theta}{\cos^2 \theta}, you can simplify it further to sin⁡2θcos⁡2θ\frac{\sin^2 \theta}{\cos^2 \theta}.

Q6: What are some related articles that I can read?

A6: Some related articles that you can read include:

Q7: What are some references that I can use?

A7: Some references that you can use include:

Conclusion

In this article, we answered some frequently asked questions (FAQs) related to the simplification of the expression csc⁡2θtan⁡2θ−1=tan⁡2θ\csc^2 \theta \tan^2 \theta - 1 = \tan^2 \theta. We provided step-by-step answers to each question, using various trigonometric identities and formulas. We also provided some related articles and references that you can use to learn more about trigonometry and simplifying expressions.

Final Answer

The final answer is sin⁡2θcos⁡2θ\boxed{\frac{\sin^2 \theta}{\cos^2 \theta}}.

References

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