Problem 74 - A Complete Resource Book in Mathematics for JEE Main

Question

The expression \(\frac{\tan A}{1-\cot A}+\frac{\cot A}{1-\tan A}\) can be written as \([\mathbf{2 0 1 3}]\) (A) \(\sec A \operatorname{cosec} A+1\) (B) \(\tan A+\cot A\) (C) \(\sec A+\operatorname{cosec} A\) (D) \(\sin A \cos A+1\)

Step-by-Step Solution

Answer

The expression simplifies to \(\sin A \cos A + 1\), which is option (D).

Step 1: Rewrite Using Trigonometric Identities

Recognize that \( \tan A = \frac{\sin A}{\cos A} \) and \( \cot A = \frac{\cos A}{\sin A} \). Use these identities to rewrite the expression:\[\frac{\tan A}{1-\cot A} = \frac{\frac{\sin A}{\cos A}}{1-\frac{\cos A}{\sin A}}\]This simplifies by calculating a common denominator for the fraction in the denominator.

Step 2: Simplify Each Term

Simplify the denominator first:\[1 - \frac{\cos A}{\sin A} = \frac{\sin A - \cos A}{\sin A}\]Substitute this back into the fraction:\[\frac{\frac{\sin A}{\cos A}}{\frac{\sin A - \cos A}{\sin A}} = \frac{\sin^2 A}{\cos A (\sin A - \cos A)}\]Similarly, simplify \( \frac{\cot A}{1 - \tan A} \):\[\frac{\frac{\cos A}{\sin A}}{\frac{\cos A - \sin A}{\cos A}} = \frac{\cos^2 A}{\sin A (\cos A - \sin A)}\]

Step 3: Combine the Fractions

Add the two simplified fractions from Step 2:\[\frac{\sin^2 A \cos A}{\cos^2 A (\sin A - \cos A)} + \frac{\cos^2 A \sin A}{\sin A (\cos A - \sin A)}\]Find a common denominator for the two terms, which is \( \sin A \cos A (\cos A - \sin A) \), to combine them:\[\frac{\sin^3 A}{\sin A \cos A (\cos A - \sin A)} + \frac{\cos^3 A}{\sin A \cos A (\cos A - \sin A)}\]

Step 4: Factor and Simplify the Numerator

Notice that both terms in the numerator have a common factor \( \sin A \cos A \):\[\frac{\sin A \cos A (\sin^2 A + \cos^2 A)}{\sin A \cos A (\cos A - \sin A)}\]Since \( \sin^2 A + \cos^2 A = 1 \):\[\frac{\sin A \cos A}{\sin A \cos A (\cos A - \sin A)} = 1\]Therefore, adding the two fractions results in:\[\sin A \cos A + 1\]

Step 5: Verify and Identify the Correct Option

Compare the final simplified expression \( \sin A \cos A + 1 \) to the given options. This matches with option \(D\): \(\sin A \cos A + 1\).

Key Concepts

Simplifying Expressions

Simplifying mathematical expressions is an essential skill in trigonometry. When tackling a problem, the goal is to break down and reduce terms to their simplest form. This often involves manipulating fractions, factoring out common terms, or applying algebraic and trigonometric identities.

In the given problem, we simplified the expression \( \frac{\tan A}{1-\cot A}+\frac{\cot A}{1-\tan A} \). Initially, this might look daunting because of its complexity. However, by expressing \( \tan A \) and \( \cot A \) using basic identities \( \tan A = \frac{\sin A}{\cos A} \) and \( \cot A = \frac{\cos A}{\sin A} \), the expression becomes more manageable.

When you tackle complex expressions, keep these tips in mind:

• Re-express terms using basic identities.
• Look for common denominators to combine fractions.
• Factor where possible to simplify numerators or denominators.

By practicing these strategies, you can turn a challenging expression into something straightforward and learn to approach trigonometric problems more efficiently.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, tangent, and their reciprocals, are foundational components in trigonometry. They allow us to relate angles to side lengths in right-angled triangles. Understanding these functions is crucial for solving trigonometric equations and simplifying expressions.

In our exercise, we start by expressing \( \tan A \) and \( \cot A \) using their respective sine and cosine identities. Understanding these relationships is pivotal:

• \( \tan A = \frac{\sin A}{\cos A} \)
• \( \cot A = \frac{\cos A}{\sin A} \)

These identities are derived from the basic definition of the tangent and cotangent functions, making them useful tools for expression simplification.

It's also important to note the relationship \( \sin^2 A + \cos^2 A = 1 \), one of the Pythagorean identities, which was key in solving the exercise. Recognizing how and when to apply these fundamental identities aids considerably in trigonometric problem-solving.

Mathematical Problem Solving

Mathematical problem solving involves a series of strategic steps to arrive at a solution, especially in the realm of trigonometry. In this exercise, the step-by-step approach was crucial to determine the simple equivalent of a complicated expression.

Start by observing the bigger picture and identifying the key identities or definitions that can be relevant. In this problem:

• Identify trigonometric identities required for simplification.
• Simplify fractions by finding common denominators.
• Factor common terms to make expressions more manageable.
• Verify your results by comparing your simplified expression with the available options.

Each step in the solution was deliberately chosen to bring the expression closer to a simpler form, demonstrating the foresight required in effective problem solving. Developing these skills helps in tackling not only specific trigonometric problems but also enhances general critical thinking and analytical abilities in mathematics.