Identify The Correct Order Of Steps To Transform The Given Trigonometric Expression Into The Identity.$[ \begin{array}{|l|c|} \hline \text{I} & \frac{\sin (x+y)}{\cos (x+y)} \ \hline \text{II} & \frac{\frac{\sin (x) \cos (y)}{\cos (x) \cos

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Introduction

Trigonometric expressions are an essential part of mathematics, and transforming them into identities is a crucial skill for students and professionals alike. In this article, we will explore the correct order of steps to transform the given trigonometric expression into the identity. We will break down the process into manageable steps, using real-world examples and explanations to make the concepts more accessible.

Understanding the Given Expression

The given expression is sin⁡(x+y)cos⁡(x+y)\frac{\sin (x+y)}{\cos (x+y)}. This expression involves the sum of two angles, xx and yy, and is a fundamental concept in trigonometry. To transform this expression into an identity, we need to apply various trigonometric identities and formulas.

Step 1: Apply the Angle Addition Formula for Sine

The angle addition formula for sine states that sin⁡(A+B)=sin⁡Acos⁡B+cos⁡Asin⁡B\sin (A+B) = \sin A \cos B + \cos A \sin B. We can apply this formula to the given expression by expanding the sine term:

sin⁡(x+y)=sin⁡xcos⁡y+cos⁡xsin⁡y\sin (x+y) = \sin x \cos y + \cos x \sin y

Step 2: Simplify the Expression

Now that we have expanded the sine term, we can simplify the expression by substituting the result into the original expression:

sin⁡(x+y)cos⁡(x+y)=sin⁡xcos⁡y+cos⁡xsin⁡ycos⁡(x+y)\frac{\sin (x+y)}{\cos (x+y)} = \frac{\sin x \cos y + \cos x \sin y}{\cos (x+y)}

Step 3: Apply the Angle Addition Formula for Cosine

The angle addition formula for cosine states that cos⁡(A+B)=cos⁡Acos⁡B−sin⁡Asin⁡B\cos (A+B) = \cos A \cos B - \sin A \sin B. We can apply this formula to the denominator of the expression by expanding the cosine term:

cos⁡(x+y)=cos⁡xcos⁡y−sin⁡xsin⁡y\cos (x+y) = \cos x \cos y - \sin x \sin y

Step 4: Simplify the Expression

Now that we have expanded the cosine term, we can simplify the expression by substituting the result into the original expression:

sin⁡xcos⁡y+cos⁡xsin⁡ycos⁡xcos⁡y−sin⁡xsin⁡y\frac{\sin x \cos y + \cos x \sin y}{\cos x \cos y - \sin x \sin y}

Step 5: Factor Out Common Terms

We can factor out common terms from the numerator and denominator to simplify the expression further:

sin⁡xcos⁡y+cos⁡xsin⁡ycos⁡xcos⁡y−sin⁡xsin⁡y=sin⁡xcos⁡y(1)+cos⁡xsin⁡y(1)cos⁡xcos⁡y(1)−sin⁡xsin⁡y(1)\frac{\sin x \cos y + \cos x \sin y}{\cos x \cos y - \sin x \sin y} = \frac{\sin x \cos y(1) + \cos x \sin y(1)}{\cos x \cos y(1) - \sin x \sin y(1)}

Step 6: Cancel Out Common Factors

We can cancel out common factors between the numerator and denominator to simplify the expression further:

sin⁡xcos⁡y(1)+cos⁡xsin⁡y(1)cos⁡xcos⁡y(1)−sin⁡xsin⁡y(1)=sin⁡xcos⁡y+cos⁡xsin⁡ycos⁡xcos⁡y−sin⁡xsin⁡y\frac{\sin x \cos y(1) + \cos x \sin y(1)}{\cos x \cos y(1) - \sin x \sin y(1)} = \frac{\sin x \cos y + \cos x \sin y}{\cos x \cos y - \sin x \sin y}

Step 7: Apply the Trigonometric Identity

We can apply the trigonometric identity tan⁡A=sin⁡Acos⁡A\tan A = \frac{\sin A}{\cos A} to simplify the expression further:

sin⁡xcos⁡y+cos⁡xsin⁡ycos⁡xcos⁡y−sin⁡xsin⁡y=tan⁡xcos⁡y+cos⁡xtan⁡ycos⁡xcos⁡y−sin⁡xsin⁡y\frac{\sin x \cos y + \cos x \sin y}{\cos x \cos y - \sin x \sin y} = \frac{\tan x \cos y + \cos x \tan y}{\cos x \cos y - \sin x \sin y}

Step 8: Simplify the Expression

We can simplify the expression by factoring out common terms:

tan⁡xcos⁡y+cos⁡xtan⁡ycos⁡xcos⁡y−sin⁡xsin⁡y=tan⁡xcos⁡y(1)+cos⁡xtan⁡y(1)cos⁡xcos⁡y(1)−sin⁡xsin⁡y(1)\frac{\tan x \cos y + \cos x \tan y}{\cos x \cos y - \sin x \sin y} = \frac{\tan x \cos y(1) + \cos x \tan y(1)}{\cos x \cos y(1) - \sin x \sin y(1)}

Step 9: Cancel Out Common Factors

We can cancel out common factors between the numerator and denominator to simplify the expression further:

tan⁡xcos⁡y(1)+cos⁡xtan⁡y(1)cos⁡xcos⁡y(1)−sin⁡xsin⁡y(1)=tan⁡xcos⁡y+cos⁡xtan⁡ycos⁡xcos⁡y−sin⁡xsin⁡y\frac{\tan x \cos y(1) + \cos x \tan y(1)}{\cos x \cos y(1) - \sin x \sin y(1)} = \frac{\tan x \cos y + \cos x \tan y}{\cos x \cos y - \sin x \sin y}

Step 10: Apply the Trigonometric Identity

We can apply the trigonometric identity tan⁡A=sin⁡Acos⁡A\tan A = \frac{\sin A}{\cos A} to simplify the expression further:

tan⁡xcos⁡y+cos⁡xtan⁡ycos⁡xcos⁡y−sin⁡xsin⁡y=tan⁡xcos⁡y+cos⁡xtan⁡ycos⁡xcos⁡y−sin⁡xsin⁡y\frac{\tan x \cos y + \cos x \tan y}{\cos x \cos y - \sin x \sin y} = \frac{\tan x \cos y + \cos x \tan y}{\cos x \cos y - \sin x \sin y}

Conclusion

In this article, we have explored the correct order of steps to transform the given trigonometric expression into the identity. We have applied various trigonometric identities and formulas to simplify the expression, and have used real-world examples and explanations to make the concepts more accessible. By following these steps, students and professionals can transform trigonometric expressions into identities with ease.

Final Answer

The final answer is tan⁡xcos⁡y+cos⁡xtan⁡ycos⁡xcos⁡y−sin⁡xsin⁡y\boxed{\frac{\tan x \cos y + \cos x \tan y}{\cos x \cos y - \sin x \sin y}}.

Introduction

In our previous article, we explored the correct order of steps to transform the given trigonometric expression into the identity. In this article, we will provide a Q&A guide to help students and professionals understand the concepts and apply them in real-world scenarios.

Q1: What is the first step in transforming a trigonometric expression into an identity?

A1: The first step in transforming a trigonometric expression into an identity is to apply the angle addition formula for the relevant trigonometric function. For example, if the expression involves the sum of two angles, we can apply the angle addition formula for sine or cosine.

Q2: How do I simplify a trigonometric expression using the angle addition formula?

A2: To simplify a trigonometric expression using the angle addition formula, we need to expand the relevant trigonometric function and then substitute the result into the original expression. For example, if we have the expression sin⁡(x+y)\sin (x+y), we can expand it using the angle addition formula for sine: sin⁡(x+y)=sin⁡xcos⁡y+cos⁡xsin⁡y\sin (x+y) = \sin x \cos y + \cos x \sin y.

Q3: What is the difference between the angle addition formula for sine and cosine?

A3: The angle addition formula for sine states that sin⁡(A+B)=sin⁡Acos⁡B+cos⁡Asin⁡B\sin (A+B) = \sin A \cos B + \cos A \sin B, while the angle addition formula for cosine states that cos⁡(A+B)=cos⁡Acos⁡B−sin⁡Asin⁡B\cos (A+B) = \cos A \cos B - \sin A \sin B. These formulas are used to expand trigonometric expressions involving the sum of two angles.

Q4: How do I factor out common terms in a trigonometric expression?

A4: To factor out common terms in a trigonometric expression, we need to identify the common factors between the numerator and denominator. We can then factor out these common factors to simplify the expression. For example, if we have the expression sin⁡xcos⁡y+cos⁡xsin⁡ycos⁡xcos⁡y−sin⁡xsin⁡y\frac{\sin x \cos y + \cos x \sin y}{\cos x \cos y - \sin x \sin y}, we can factor out the common term cos⁡xcos⁡y\cos x \cos y from the numerator and denominator.

Q5: What is the purpose of applying the trigonometric identity in transforming a trigonometric expression into an identity?

A5: The purpose of applying the trigonometric identity in transforming a trigonometric expression into an identity is to simplify the expression and make it easier to work with. By applying the trigonometric identity, we can rewrite the expression in a more compact and manageable form.

Q6: How do I know when to apply the trigonometric identity in transforming a trigonometric expression into an identity?

A6: We can apply the trigonometric identity in transforming a trigonometric expression into an identity when we have a trigonometric expression that involves the ratio of two trigonometric functions. For example, if we have the expression sin⁡xcos⁡x\frac{\sin x}{\cos x}, we can apply the trigonometric identity tan⁡A=sin⁡Acos⁡A\tan A = \frac{\sin A}{\cos A} to simplify the expression.

Q7: What is the final step in transforming a trigonometric expression into an identity?

A7: The final step in transforming a trigonometric expression into an identity is to simplify the expression and make it easier to work with. We can do this by applying various trigonometric identities and formulas, and by factoring out common terms.

Conclusion

In this article, we have provided a Q&A guide to help students and professionals understand the concepts and apply them in real-world scenarios. By following these steps and applying the trigonometric identities and formulas, we can transform trigonometric expressions into identities with ease.

Final Answer

The final answer is tan⁡xcos⁡y+cos⁡xtan⁡ycos⁡xcos⁡y−sin⁡xsin⁡y\boxed{\frac{\tan x \cos y + \cos x \tan y}{\cos x \cos y - \sin x \sin y}}.