Deriving the Desired Expression

Oct 13, 2025 by windailys.com 62 views

Introduction

Trigonometric equations are a fundamental concept in mathematics, and solving them requires a deep understanding of trigonometric functions and their properties. In this article, we will explore a specific trigonometric equation involving cosine and sine functions, and demonstrate how to solve it using algebraic manipulations and trigonometric identities.

The Given Equations

We are given two equations involving cosine and sine functions:

\cos A - \sin A = m

\cos A + \sin A = n

Our objective is to show that:

\frac{m^2 - n^2}{m^2 + n^2} = -2 \sin A \cdot \cos A = -\frac{2}{\tan A + \cot A}

Solving the Equations

To solve the given equations, we can start by squaring both equations:

\cos^2 A - 2 \sin A \cos A + \sin^2 A = m^2

\cos^2 A + 2 \sin A \cos A + \sin^2 A = n^2

Now, we can add the two equations to eliminate the $\sin^2 A$ term:

2 \cos^2 A = m^2 + n^2

\cos^2 A = \frac{m^2 + n^2}{2}

Next, we can subtract the first equation from the second equation to eliminate the $\cos^2 A$ term:

4 \sin A \cos A = n^2 - m^2

2 \sin A \cos A = \frac{n^2 - m^2}{2}

Deriving the Desired Expression

Now, we can substitute the expressions for $\cos^2 A$ and $2 \sin A \cos A$ into the desired expression:

\frac{m^2 - n^2}{m^2 + n^2} = -2 \sin A \cos A

\frac{m^2 - n^2}{m^2 + n^2} = -\frac{n^2 - m^2}{m^2 + n^2}

\frac{m^2 - n^2}{m^2 + n^2} = -\frac{2 \sin A \cos A}{\cos^2 A}

\frac{m^2 - n^2}{m^2 + n^2} = -2 \tan A

Using Trigonometric Identities

We can also use trigonometric identities to derive the desired expression:

\frac{m^2 - n^2}{m^2 + n^2} = -2 \sin A \cos A

\frac{m^2 - n^2}{m^2 + n^2} = -\frac{2}{\frac{\sin A}{\cos A} + \frac{\cos A}{\sin A}}

\frac{m^2 - n^2}{m^2 + n^2} = -\frac{2}{\tan A + \cot A}

Conclusion

In this article, we have demonstrated how to solve a trigonometric equation involving cosine and sine functions, and derived the desired expression using algebraic manipulations and trigonometric identities. The key steps involved squaring both equations, adding and subtracting them to eliminate terms, and using trigonometric identities to derive the final expression. This problem requires a deep understanding of trigonometric functions and their properties, and is a great example of how to solve complex trigonometric equations.

Key Takeaways

Squaring both equations can help eliminate terms and simplify the equation.
Adding and subtracting the equations can help eliminate terms and simplify the equation.
Trigonometric identities can be used to derive the final expression.
This problem requires a deep understanding of trigonometric functions and their properties.

References

CBSE 20.
Q&A: Trigonometric Equations and Identities =============================================

Introduction

In our previous article, we explored a trigonometric equation involving cosine and sine functions, and demonstrated how to solve it using algebraic manipulations and trigonometric identities. In this article, we will answer some common questions related to trigonometric equations and identities.

Q: What is a trigonometric equation?

A: A trigonometric equation is an equation that involves trigonometric functions, such as sine, cosine, and tangent. These equations can be used to model real-world problems, such as the motion of objects, the behavior of waves, and the properties of triangles.

Q: How do I solve a trigonometric equation?

A: To solve a trigonometric equation, you can use algebraic manipulations, such as squaring both sides of the equation, adding and subtracting terms, and using trigonometric identities. You can also use trigonometric tables or calculators to find the values of trigonometric functions.

Q: What is a trigonometric identity?

A: A trigonometric identity is a statement that two or more trigonometric expressions are equal. These identities can be used to simplify trigonometric equations and to find the values of trigonometric functions.

Q: How do I use trigonometric identities to solve equations?

A: To use trigonometric identities to solve equations, you can start by identifying the trigonometric functions involved in the equation. Then, you can use the relevant trigonometric identity to simplify the equation and find the values of the trigonometric functions.

Q: What are some common trigonometric identities?

A: Some common trigonometric identities include:

$\sin^2 A + \cos^2 A = 1$
$\tan A = \frac{\sin A}{\cos A}$
$\cot A = \frac{\cos A}{\sin A}$
$\sec A = \frac{1}{\cos A}$
$\csc A = \frac{1}{\sin A}$

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

A: To use trigonometric identities to simplify expressions, you can start by identifying the trigonometric functions involved in the expression. Then, you can use the relevant trigonometric identity to simplify the expression and find the values of the trigonometric functions.

Q: What are some common applications of trigonometric equations and identities?

A: Some common applications of trigonometric equations and identities include:

Modeling the motion of objects
Analyzing the behavior of waves
Studying the properties of triangles
Solving problems in physics, engineering, and computer science

Conclusion

In this article, we have answered some common questions related to trigonometric equations and identities. We hope that this article has provided you with a better understanding of these concepts and how to use them to solve problems.

Key Takeaways

Trigonometric equations and identities are used to model real-world problems and to solve equations.
Algebraic manipulations and trigonometric identities can be used to solve trigonometric equations.
Trigonometric identities can be used to simplify expressions and to find the values of trigonometric functions.
Trigonometric equations and identities have many applications in physics, engineering, and computer science.

References

CBSE 20.