Problem 74 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The expression \(\frac{\tan A}{1-\cot A}+\frac{\cot A}{1-\tan A}\) can be written as (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 (C) \( \sec A + \operatorname{cosec} A \).

Step 1: Understand the Problem

We have the expression \( \frac{\tan A}{1-\cot A}+\frac{\cot A}{1-\tan A} \) and need to simplify it. The goal is to rewrite it in one of given forms: (A) \( \sec A \operatorname{cosec} A+1 \), (B) \( \tan A+\cot A \), (C) \( \sec A+\operatorname{cosec} A \), or (D) \( \sin A \cos A+1 \).

Step 2: Apply Trigonometric Identities

Recall that \( \cot A = \frac{1}{\tan A} \), so \( 1-\cot A = 1-\frac{1}{\tan A} = \frac{\tan A - 1}{\tan A} \). Similarly, \( 1-\tan A \) remains as it is. This gives us a clearer form for the denominators.

Step 3: Rewrite the Fractions

We rewrite the original expression as: \( \frac{\tan A}{\frac{\tan A - 1}{\tan A}} + \frac{\cot A}{1-\tan A} \). Simplifying the first fraction gives \( \frac{\tan^2 A}{\tan A - 1} \). The second fraction remains as \( \frac{\cot A}{1-\tan A} \).

Step 4: Combine the Fractions

Rewrite \( \frac{\tan^2 A}{\tan A - 1} + \frac{1/\tan A}{1-\tan A} \) as \( \frac{\tan^2 A}{\tan A - 1} + \frac{1}{\tan A (1-\tan A)} \). Factor and find common denominators.

Step 5: Simplify the Expression

Combine the numerators over the common denominator. The combined numerator becomes \( \tan^3 A + 1 \) and the common denominator is \((\tan A - 1)(1-\tan A)\), simplifying using \( a^3 + b^3 = (a+b)(a^2-ab+b^2) \).

Step 6: Match with Options

Through simplifications, identify how the terms correspond to the choices given in the problem. Simplify the expression to the form \( \sec A + \operatorname{cosec} A \), matching option (C).

Key Concepts

Trigonometric Simplification

When dealing with trigonometric expressions, sometimes they appear to be complicated at first glance. Simplifying these expressions is a process that involves using trigonometric identities and algebraic manipulation to rewrite them in a simpler form. In the exercise, we're given a compound fraction: \[ \frac{\tan A}{1-\cot A} + \frac{\cot A}{1-\tan A} \]The first step in simplification is to apply basic trigonometric identities. Here, noting that \( \cot A = \frac{1}{\tan A} \), we can rewrite parts of the expression in terms of \( \tan A \). This approach helps reveal common factors or terms that can simplify the expression. Remember:

• Trigonometric identities like \( \cot A = \frac{1}{\tan A} \) allow interchangeability between functions that seem distinct.
• Simplifying involves rewriting expressions in terms of fewer trigonometric functions where possible.

Breaking down fractions often means finding a common denominator, which is key in bringing multiple fractions together into a cohesive, simplified expression. The goal is always to reach a form that either matches a target form or simplifies complex numerators and denominators.

Trigonometric Expressions

A trigonometric expression is any math statement that involves trigonometric functions like sine, cosine, tangent, and their reciprocals. In the given problem, the expression involves tangent \( \tan A \) and cotangent \( \cot A \).
Trigonometric expressions may look complicated initially, but their true nature often comes to light when you:

• Utilize known identities.
• Apply algebraic techniques to simplify the terms.

In our question, we've used identities to transform \( \frac{\tan A}{1-\cot A} + \frac{\cot A}{1-\tan A} \) into a simpler form. Realizing that the expression involves both fractions and trigonometric functions, a strategic approach involves:

• Rewriting functions to improve manipulability, like using \( \cot A = \frac{1}{\tan A} \).
• Finding a common denominator to combine fractions efficiently.

Ultimately, understanding and manipulating trigonometric expressions are about recognizing patterns and applying identities to simplify and transform complex equations into recognizable, manageable forms.

Angle Transformations

In trigonometry, angle transformations involve changing the form of trigonometric functions given angles, often to simplify calculations or expressions. In our problem, although the angle \( A \) itself is not transformed directly, the expression makes use of reciprocal identities which themselves are a type of transformation.Understanding these transformations helps us turn functions like \( \tan A \) and \( \cot A \) into each other or into simpler terms that can be easily managed. When simplifying expressions or solving problems, such transformations might involve:

• Switching between \( \tan A \) and \( \cot A \) using \( \cot A = \frac{1}{\tan A} \).
• Using identities such as \( \tan^2 A + 1 = \sec^2 A \) to relate angles and their functions.

These techniques allow for the expression to be rearranged and simplified, making problem-solving more efficient. Transformations are about finding the best pathway from a complex starting point to a simplified, solved conclusion. It's not just about changing the form, but also about recognizing which transformations can lead efficiently to desired results, such as in this case, reaching the form \( \sec A + \operatorname{cosec A} \).