Problem 2 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(\mathrm{A}=\cot ^{-1} \sqrt{\tan \theta}-\tan ^{-1} \sqrt{\tan \theta}\), then \(\tan \left(\frac{\pi}{4}-\frac{\mathrm{A}}{2}\right)\) is equal to (A) \(\sqrt{\cot \theta} \theta\) (B) \(\tan \theta\) (C) \(\sqrt{\tan \theta}\) (D) none of these

Step-by-Step Solution

Answer

\(\sqrt{\tan \theta}\) (Option C)

Step 1: Simplify the Expression for A

The given expression is \(A = \cot^{-1}(\sqrt{\tan \theta}) - \tan^{-1}(\sqrt{\tan \theta})\). We recognize that for any \(x\), \(\cot^{-1}(x) + \tan^{-1}(x) = \frac{\pi}{2}\). Therefore, we can deduce that \(\cot^{-1}(\sqrt{\tan \theta}) = \frac{\pi}{2} - \tan^{-1}(\sqrt{\tan \theta})\). Substitute back into A: \[A = \left( \frac{\pi}{2} - \tan^{-1}(\sqrt{\tan \theta}) \right) - \tan^{-1}(\sqrt{\tan \theta})\]\[A = \frac{\pi}{2} - 2 \tan^{-1}(\sqrt{\tan \theta})\].

Step 2: Use the Identity for Tan Function

Our goal is to evaluate \(\tan \left(\frac{\pi}{4} - \frac{A}{2}\right)\). Substitute the expression for \(A\) from Step 1:\[\frac{A}{2} = \frac{\pi}{4} - \tan^{-1}(\sqrt{\tan \theta})\]Then, our expression becomes:\[\tan \left(\frac{\pi}{4} - \left(\frac{\pi}{4} - \tan^{-1}(\sqrt{\tan \theta})\right)\right) = \tan \left(\tan^{-1}(\sqrt{\tan \theta})\right) = \sqrt{\tan \theta}\]

Step 3: Compare with Given Answers

The result from Step 2 is \(\sqrt{\tan \theta}\). Looking at the multiple-choice options:- A) \(\sqrt{\cot \theta}\)- B) \(\tan \theta\)- C) \(\sqrt{\tan \theta}\)- D) None of theseThe correct answer matches option C.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are the reverse operations of the basic trigonometric functions. They help in finding angles when the value of the trigonometric function is known. For example, if we know the value of \( \tan \theta \), we can use the inverse tangent function, \( \tan^{-1}(x) \), to find the angle \( \theta \). This makes inverse trigonometric functions particularly useful in solving equations and finding unknown angles in triangles.
In the given exercise, inverse trigonometric functions \( \cot^{-1}(x) \) and \( \tan^{-1}(x) \) were key to simplifying the expression for \( A \). An interesting property used was \( \cot^{-1}(x) + \tan^{-1}(x) = \frac{\pi}{2} \), which simplifies expressions by turning complex combinations into more solvable forms.

• Key functions: \( \tan^{-1}(x), \cot^{-1}(x) \)
• Useful properties: \( \cot^{-1}(x) + \tan^{-1}(x) = \frac{\pi}{2} \)

This simplification is crucial in solving such mathematical problems efficiently.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are always true for every value of the occurring variables where the functions are defined. These identities are useful tools in simplifying expressions and solving trigonometric equations.
In the problem provided, the identity \( \cot^{-1}(x) + \tan^{-1}(x) = \frac{\pi}{2} \) was used to simplify the expression for \( A \). Moreover, we utilized the tangent subtraction formula to find \( \tan \left(\frac{\pi}{4} - \frac{A}{2}\right) \).

• Useful identities: \( \cot^{-1}(x) + \tan^{-1}(x) = \frac{\pi}{2} \)
• Trigonometric subtraction: \( \tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b} \)

Trigonometric identities help transform complicated expressions into manageable formulas, making problem-solving easier.

Mathematical Problem Solving

Mathematical problem solving is a skill that involves understanding the problem, devising a plan, carrying out the plan, and evaluating the solution for accuracy. When solving trigonometric problems, having a strategic approach can significantly improve efficiency and accuracy.
Let's break down the steps seen in the problem:

• Understanding the Problem: Initially, recognize the need to simplify \( A \) using inverse functions.
• Planning: Use known identities to simplify \( A \) and solve for \( \tan \left(\frac{\pi}{4} - \frac{A}{2}\right) \).
• Execution: Substitute and apply trigonometric identities for simplification.
• Evaluation: Match the solution with multiple-choice options to find the correct answer.

Approaching problems methodically ensures that each step is followed attentively and correctly, reducing errors and leading to the correct solution. Effective problem solving requires not only mathematical skills but also creativity and persistence in applying the right strategies.