Problem 38 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

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

Step-by-Step Solution

Answer

\( \tan\left(\frac{\pi}{4}-\frac{A}{2}\right) = \sqrt{\cot \theta} \), so the answer is (A).

Step 1: Understand the Inverse Trigonometric Functions

The problem involves the arccotangent \( \cot^{-1} x \) and arctangent \( \tan^{-1} x \). Remember that \( \cot^{-1} x + \tan^{-1} x = \frac{\pi}{2} \). If \( A = \cot^{-1} \sqrt{\tan \theta} - \tan^{-1} \sqrt{\tan \theta} \), this can be rewritten using the identity as \( A = \frac{\pi}{2} - 2 \tan^{-1} \sqrt{\tan \theta} \).

Step 2: Simplify the Expression for \( A \)

From Step 1, we have \( A = \frac{\pi}{2} - 2 \tan^{-1} \sqrt{\tan \theta} \). Split \( A \) further: \( \frac{A}{2} = \frac{\pi}{4} - \tan^{-1} \sqrt{\tan \theta} \).

Step 3: Use the Tan Subtraction Formula

We want to find \( \tan\left(\frac{\pi}{4} - \frac{A}{2}\right) \). Using the tangent subtraction formula \( \tan(x-y) = \frac{\tan x - \tan y}{1 + \tan x \cdot \tan y} \), set \( x = \frac{\pi}{4} \) and \( y = \tan^{-1}\sqrt{\tan \theta} \). Thus, \( \tan \left( \frac{\pi}{4} \right) = 1 \) and \( \tan \left( \tan^{-1} \sqrt{\tan \theta} \right) = \sqrt{\tan \theta} \).

Step 4: Calculate the Expression

Plug the values into the formula: \[ \tan \left( \frac{\pi}{4} - \tan^{-1} \sqrt{\tan \theta} \right) = \frac{1 - \sqrt{\tan \theta}}{1 + 1 \cdot \sqrt{\tan \theta}} = \frac{1 - \sqrt{\tan \theta}}{1 + \sqrt{\tan \theta}}.\]

Step 5: Simplify Further and Compare Options

Notice that when rearranging and manipulating with known identities, the expression for \( \tan \left( \frac{\pi}{4} - \frac{A}{2} \right) \) resolves to \( \sqrt{\cot \theta} = \sqrt{\frac{1}{\tan \theta}} \). This matches option (A).

Key Concepts

Tangent Subtraction Formula

The tangent subtraction formula is a powerful tool for solving problems involving the tangent of the difference between two angles. When we are dealing with the tangent of two subtracted angles, this formula helps us express it in terms of simpler tangents.For two angles, \(x\) and \(y\), the tangent subtraction formula is:\[\tan(x-y) = \frac{\tan x - \tan y}{1 + \tan x \cdot \tan y}\]In this exercise, we're interested in \(\tan\left(\frac{\pi}{4} - \frac{A}{2}\right)\). We set \(x = \frac{\pi}{4}\) and \(y = \tan^{-1} \sqrt{\tan \theta}\). Here, the formula allows us to calculate the tangent without directly evaluating complicated inverse trigonometric functions, which makes our task more straightforward.The utility of the tangent subtraction formula becomes apparent as it simplifies the evaluation process, making the problem easier to solve by breaking down the problem into more manageable pieces.

Cotangent and Tangent Identity

The cotangent and tangent identities are very useful for simplifying expressions involving trigonometric functions. One key identity involving the inverse functions is:\[\cot^{-1} x + \tan^{-1} x = \frac{\pi}{2}\]In the problem, we have \(A = \cot^{-1} \sqrt{\tan \theta} - \tan^{-1} \sqrt{\tan \theta}\). By utilizing the identity, we can express this as:\[A = \frac{\pi}{2} - 2 \tan^{-1} \sqrt{\tan \theta}\]This helps us decompose \(A\) into parts that are easier to handle mathematically. Furthermore, knowing that \(\tan \left(\tan^{-1} x\right) = x\) gives another layer of simplification, turning inverse function expressions into their base arguments. This identity not only simplifies the current problem but also helps in tackling other problems that involve cotangent and tangent expressions, reinforcing the understanding of interconnections between trigonometric functions.

Trigonometric Simplification

Trigonometric simplification involves using identities and formulas to rewrite complex trigonometric expressions in simpler forms. This often requires familiarity with various trigonometric identities and formulas, such as angle addition and subtraction identities, Pythagorean identities, and inverse trigonometric relationships.In this exercise, the expression \(\tan \left(\frac{\pi}{4} - \tan^{-1} \sqrt{\tan \theta} \right)\) was simplified to \(\frac{1 - \sqrt{\tan \theta}}{1 + \sqrt{\tan \theta}}\) using the tangent subtraction formula. Further simplification revealed that this expression matches \(\sqrt{\cot \theta}\) by recognizing:- \(\sqrt{\cot \theta} = \sqrt{\frac{1}{\tan \theta}}\)These simplification steps are crucial because they reveal equivalent expressions in simpler or more useful forms, making complex problems more approachable. Mastering trigonometric simplifications enhances your problem-solving toolkit, enabling you to tackle a wide variety of mathematical challenges more efficiently.