(a) Prove That: Tan ⁡ Θ 1 − Cot ⁡ Θ + Cot ⁡ Θ 1 − Tan ⁡ Θ = 1 + Sec ⁡ Θ Cosec ⁡ Θ \frac{\tan \theta}{1-\cot \theta}+\frac{\cot \theta}{1-\tan \theta}=1+\sec \theta \operatorname{cosec} \theta 1 − C O T Θ T A N Θ ​ + 1 − T A N Θ C O T Θ ​ = 1 + Sec Θ Cosec Θ

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Introduction

In this article, we will delve into the world of trigonometry and explore a fascinating identity involving tangent and cotangent functions. The given expression is a sum of two fractions, each containing these trigonometric functions. Our goal is to simplify this expression and prove that it is equal to a specific combination of secant and cosecant functions.

Understanding the Trigonometric Functions

Before we begin, let's take a moment to understand the trigonometric functions involved in this problem. The tangent function, denoted by $\tan \theta$, is defined as the ratio of the sine and cosine functions: $\tan \theta = \frac{\sin \theta}{\cos \theta}$. Similarly, the cotangent function, denoted by $\cot \theta$, is defined as the ratio of the cosine and sine functions: $\cot \theta = \frac{\cos \theta}{\sin \theta}$.

Simplifying the Expression

To simplify the given expression, we will start by finding a common denominator for the two fractions. The common denominator will be $(1-\cot \theta)(1-\tan \theta)$. We can then rewrite each fraction with this common denominator.

\frac{\tan \theta}{1-\cot \theta}+\frac{\cot \theta}{1-\tan \theta} = \frac{\tan \theta(1-\tan \theta)}{(1-\cot \theta)(1-\tan \theta)} + \frac{\cot \theta(1-\cot \theta)}{(1-\tan \theta)(1-\cot \theta)}

Expanding and Simplifying

Now, let's expand and simplify each fraction.

\frac{\tan \theta(1-\tan \theta)}{(1-\cot \theta)(1-\tan \theta)} = \frac{\tan \theta - \tan^2 \theta}{1-\cot \theta - \tan \theta + \tan \cot \theta}

\frac{\cot \theta(1-\cot \theta)}{(1-\tan \theta)(1-\cot \theta)} = \frac{\cot \theta - \cot^2 \theta}{1-\tan \theta - \cot \theta + \cot \tan \theta}

Combining the Fractions

Now, let's combine the two fractions by adding them together.

\frac{\tan \theta - \tan^2 \theta}{1-\cot \theta - \tan \theta + \tan \cot \theta} + \frac{\cot \theta - \cot^2 \theta}{1-\tan \theta - \cot \theta + \cot \tan \theta}

Simplifying the Combined Expression

To simplify the combined expression, we can start by combining the numerators.

\frac{\tan \theta - \tan^2 \theta + \cot \theta - \cot^2 \theta}{1-\cot \theta - \tan \theta + \tan \cot \theta + 1-\tan \theta - \cot \theta + \cot \tan \theta}

Canceling Common Terms

Now, let's cancel out any common terms in the numerator and denominator.

\frac{\tan \theta - \tan^2 \theta + \cot \theta - \cot^2 \theta}{2 - 2 \tan \theta - 2 \cot \theta + \tan \cot \theta + \cot \tan \theta}

Factoring the Numerator

To simplify the numerator further, let's factor it.

\frac{(\tan \theta - \cot \theta)(1 - \tan \theta \cot \theta)}{2 - 2 \tan \theta - 2 \cot \theta + \tan \cot \theta + \cot \tan \theta}

Simplifying the Denominator

Now, let's simplify the denominator by combining like terms.

\frac{(\tan \theta - \cot \theta)(1 - \tan \theta \cot \theta)}{2(1 - \tan \theta)(1 - \cot \theta)}

Canceling Common Terms

Now, let's cancel out any common terms in the numerator and denominator.

\frac{\tan \theta - \cot \theta}{2(1 - \tan \theta)(1 - \cot \theta)}

Simplifying the Expression

To simplify the expression further, let's multiply the numerator and denominator by $-1$.

\frac{\cot \theta - \tan \theta}{2(1 - \tan \theta)(1 - \cot \theta)}

Simplifying the Expression

To simplify the expression further, let's multiply the numerator and denominator by $\frac{1}{\sin \theta \cos \theta}$.

\frac{\frac{\cot \theta - \tan \theta}{\sin \theta \cos \theta}}{\frac{2(1 - \tan \theta)(1 - \cot \theta)}{\sin \theta \cos \theta}}

Simplifying the Expression

To simplify the expression further, let's simplify the numerator and denominator separately.

\frac{\frac{\frac{\cos \theta}{\sin \theta} - \frac{\sin \theta}{\cos \theta}}{\sin \theta \cos \theta}}{\frac{2(1 - \frac{\sin \theta}{\cos \theta})(1 - \frac{\cos \theta}{\sin \theta})}{\sin \theta \cos \theta}}

Simplifying the Expression

To simplify the expression further, let's simplify the numerator and denominator separately.

\frac{\frac{\cos^2 \theta - \sin^2 \theta}{\sin \theta \cos \theta}}{\frac{2(\cos^2 \theta - \sin^2 \theta)}{\sin \theta \cos \theta}}

Simplifying the Expression

To simplify the expression further, let's simplify the numerator and denominator separately.

\frac{\cos^2 \theta - \sin^2 \theta}{2(\cos^2 \theta - \sin^2 \theta)}

Canceling Common Terms

Now, let's cancel out any common terms in the numerator and denominator.

\frac{1}{2}

Conclusion

In this article, we have successfully simplified the given expression and proved that it is equal to $\frac{1}{2}$. This result is a fascinating example of the power of trigonometric identities and the importance of simplifying complex expressions.

Final Answer

The final answer is $\boxed{\frac{1}{2}}$.

Introduction

In our previous article, we explored a fascinating trigonometric identity involving tangent and cotangent functions. We successfully simplified the given expression and proved that it is equal to $\frac{1}{2}$. In this article, we will answer some common questions related to this identity and provide additional insights into the world of trigonometry.

Q: What is the significance of this trigonometric identity?

A: This identity is significant because it demonstrates the power of trigonometric functions and their ability to simplify complex expressions. It also highlights the importance of understanding the relationships between different trigonometric functions.

Q: How can I apply this identity in real-world problems?

A: This identity can be applied in various real-world problems, such as physics, engineering, and navigation. For example, it can be used to calculate the height of a building or the distance between two objects.

Q: What are some common mistakes to avoid when simplifying trigonometric expressions?

A: Some common mistakes to avoid when simplifying trigonometric expressions include:

• Not canceling out common terms
• Not using the correct trigonometric identities
• Not simplifying the expression enough

Q: How can I prove this identity using different methods?

A: There are several methods to prove this identity, including:

• Using the definition of tangent and cotangent functions
• Using the Pythagorean identity
• Using the sum and difference formulas

Q: What are some other trigonometric identities that I should know?

A: Some other trigonometric identities that you should know include:

• The Pythagorean identity: $\sin^2 \theta + \cos^2 \theta = 1$
• The sum and difference formulas: $\sin (A + B) = \sin A \cos B + \cos A \sin B$ and $\cos (A + B) = \cos A \cos B - \sin A \sin B$
• The double-angle formulas: $\sin 2\theta = 2\sin \theta \cos \theta$ and $\cos 2\theta = \cos^2 \theta - \sin^2 \theta$

Q: How can I practice and improve my skills in trigonometry?

A: To practice and improve your skills in trigonometry, you can:

• Work on solving problems and exercises
• Practice simplifying trigonometric expressions
• Use online resources and tutorials
• Join a study group or find a study partner

Conclusion

In this article, we have answered some common questions related to the trigonometric identity and provided additional insights into the world of trigonometry. We hope that this article has been helpful in clarifying any doubts and providing a better understanding of this fascinating topic.

Final Answer

The final answer is $\boxed{\frac{1}{2}}$.