Simplify To A Single Trigonometric Function With No Denominator: Tan ⁡ 2 Θ Sin ⁡ 2 Θ \frac{\tan^2 \theta}{\sin^2 \theta} S I N 2 Θ T A N 2 Θ ​ Answer: □ \square □ Θ \theta Θ

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Simplify to a Single Trigonometric Function with No Denominator: tan⁡2θsin⁡2θ\frac{\tan^2 \theta}{\sin^2 \theta}

In this article, we will explore the process of simplifying a given trigonometric expression to a single function with no denominator. The expression we will be working with is tan⁡2θsin⁡2θ\frac{\tan^2 \theta}{\sin^2 \theta}. We will use various trigonometric identities and properties to simplify this expression and arrive at a final answer.

Understanding the Given Expression

The given expression is tan⁡2θsin⁡2θ\frac{\tan^2 \theta}{\sin^2 \theta}. To simplify this expression, we need to understand the properties of the trigonometric functions involved. The tangent function is defined as the ratio of the sine and cosine functions, i.e., tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}. Similarly, the sine function is defined as the ratio of the opposite side to the hypotenuse in a right-angled triangle.

Step 1: Simplify the Expression Using Trigonometric Identities

To simplify the given expression, we can start by using the trigonometric identity tan⁡2θ=sin⁡2θcos⁡2θ\tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta}. This identity allows us to rewrite the numerator of the given expression in terms of the sine and cosine functions.

import sympy as sp

theta = sp.symbols('theta')



expr = (sp.tan(theta)**2) / (sp.sin(theta)**2)



simplified_expr = expr.subs(sp.tan(theta)**2, sp.sin(theta)**2/sp.cos(theta)**2)


print(simplified_expr)

Step 2: Cancel Out Common Factors

After simplifying the expression using the trigonometric identity, we can cancel out common factors in the numerator and denominator. In this case, we can cancel out the sin⁡2θ\sin^2 \theta term in the numerator and denominator.

# Cancel out common factors
canceled_expr = simplified_expr.cancel()

print(canceled_expr)

Step 3: Simplify the Expression Further

After canceling out common factors, we can simplify the expression further by using the trigonometric identity cos⁡2θ=1−sin⁡2θ\cos^2 \theta = 1 - \sin^2 \theta. This identity allows us to rewrite the denominator of the expression in terms of the sine function.

# Simplify the expression further using the trigonometric identity
further_simplified_expr = canceled_expr.subs(sp.cos(theta)**2, 1-sp.sin(theta)**2)

print(further_simplified_expr)

Step 4: Final Answer

After simplifying the expression using the trigonometric identities and canceling out common factors, we can arrive at the final answer.

# Final answer
final_answer = further_simplified_expr

print(final_answer)

In this article, we simplified the given trigonometric expression tan⁡2θsin⁡2θ\frac{\tan^2 \theta}{\sin^2 \theta} to a single function with no denominator. We used various trigonometric identities and properties to simplify the expression and arrive at the final answer. The final answer is sec⁡2θ\boxed{\sec^2 \theta}.

sec⁡2θ\boxed{\sec^2 \theta}
Simplify to a Single Trigonometric Function with No Denominator: tan⁡2θsin⁡2θ\frac{\tan^2 \theta}{\sin^2 \theta} - Q&A

In our previous article, we simplified the given trigonometric expression tan⁡2θsin⁡2θ\frac{\tan^2 \theta}{\sin^2 \theta} to a single function with no denominator. We used various trigonometric identities and properties to arrive at the final answer. In this article, we will answer some frequently asked questions related to the simplification of this expression.

Q: What is the tangent function?

A: The tangent function is defined as the ratio of the sine and cosine functions, i.e., tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}.

Q: What is the sine function?

A: The sine function is defined as the ratio of the opposite side to the hypotenuse in a right-angled triangle.

Q: How do I simplify the expression tan⁡2θsin⁡2θ\frac{\tan^2 \theta}{\sin^2 \theta}?

A: To simplify the expression, you can start by using the trigonometric identity tan⁡2θ=sin⁡2θcos⁡2θ\tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta}. This identity allows you to rewrite the numerator of the expression in terms of the sine and cosine functions. Then, you can cancel out common factors in the numerator and denominator.

Q: What is the final answer to the expression tan⁡2θsin⁡2θ\frac{\tan^2 \theta}{\sin^2 \theta}?

A: The final answer to the expression is sec⁡2θ\boxed{\sec^2 \theta}.

Q: What is the secant function?

A: The secant function is defined as the reciprocal of the cosine function, i.e., sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta}.

Q: How do I use the trigonometric identities to simplify the expression?

A: To use the trigonometric identities to simplify the expression, you can start by rewriting the numerator and denominator of the expression in terms of the sine and cosine functions. Then, you can cancel out common factors and use the trigonometric identities to simplify the expression further.

Q: What are some common trigonometric identities?

A: Some common trigonometric identities include:

  • sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1
  • tan⁡2θ=sin⁡2θcos⁡2θ\tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta}
  • sec⁡2θ=1cos⁡2θ\sec^2 \theta = \frac{1}{\cos^2 \theta}

Q: How do I know which trigonometric identity to use?

A: To know which trigonometric identity to use, you need to analyze the expression and identify the trigonometric functions involved. Then, you can use the trigonometric identities to rewrite the expression in a simpler form.

In this article, we answered some frequently asked questions related to the simplification of the expression tan⁡2θsin⁡2θ\frac{\tan^2 \theta}{\sin^2 \theta}. We provided explanations and examples to help you understand the trigonometric identities and how to use them to simplify the expression. We hope this article has been helpful in clarifying any doubts you may have had.

sec⁡2θ\boxed{\sec^2 \theta}