Problem 2 - A Complete Resource Book in Mathematics for JEE Main 2017

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

The correct answer is (C) \( \sqrt{\tan \theta} \).

Step 1: Understand the Problem

We need to find the value of \( \tan \left(\frac{\pi}{4}-\frac{\mathrm{A}}{2}\right) \), where \( \mathrm{A} = \cot^{-1} \sqrt{\tan \theta} - \tan^{-1} \sqrt{\tan \theta} \). We have to evaluate what \( \mathrm{A} \) simplifies to and then substitute it in the expression.

Step 2: Simplify the Expression for A

Let's simplify the expression \( \mathrm{A} = \cot^{-1} \sqrt{\tan \theta} - \tan^{-1} \sqrt{\tan \theta} \). Using the identity \( \cot^{-1} x = \frac{\pi}{2} - \tan^{-1} x \), we can express the cotangent inverse in terms of tangent inverse: \[ \cot^{-1} \sqrt{\tan \theta} = \frac{\pi}{2} - \tan^{-1} \sqrt{\tan \theta} \]Thus, \( \mathrm{A} = \left(\frac{\pi}{2} - \tan^{-1} \sqrt{\tan \theta}\right) - \tan^{-1} \sqrt{\tan \theta} \), which simplifies to \( \mathrm{A} = \frac{\pi}{2} - 2\tan^{-1} \sqrt{\tan \theta} \).

Step 3: Find A/2

Now let's find the value of \( \mathrm{A}/2 \):\[ \frac{\mathrm{A}}{2} = \frac{1}{2} \left( \frac{\pi}{2} - 2\tan^{-1} \sqrt{\tan \theta} \right) \]Thus, \[ \frac{\mathrm{A}}{2} = \frac{\pi}{4} - \tan^{-1} \sqrt{\tan \theta} \].

Step 4: Simplify the Target Expression

Substitute \( \frac{\mathrm{A}}{2} \) into the expression \( \tan \left(\frac{\pi}{4} - \frac{\mathrm{A}}{2}\right) \), giving us:\[ \tan \left(\frac{\pi}{4} - \left(\frac{\pi}{4} - \tan^{-1} \sqrt{\tan \theta}\right) \right) = \tan \left(\tan^{-1} \sqrt{\tan \theta} \right) \]Since \( \tan \left(\tan^{-1} x \right) = x \), the expression simplifies to \( \sqrt{\tan \theta} \).

Step 5: Select the Correct Answer

Comparing \( \sqrt{\tan \theta} \) with the given options, option (C) is \( \sqrt{\tan \theta} \). Thus, (C) is the correct answer.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every angle. Understanding these identities is crucial because they assist in simplifying complex trigonometric problems. In the context of inverse trigonometric functions, one of the key identities is the relationship between inverse tangent and inverse cotangent functions:

• \( \cot^{-1}(x) = \frac{\pi}{2} - \tan^{-1}(x) \)

This identity becomes particularly useful when you need to simplify or transform expressions involving these inverses, as seen in the original problem. It essentially converts cotangent expressions into tangent expressions, thereby reducing the complexity when calculations are done.
With practice, you can quickly recognize and apply these identities to find quicker solutions to trigonometric problems.

Angle Transformation

Angle transformation refers to the process of altering the angle expression to simplify trigonometric equations or expressions. When dealing with inverse trigonometric functions, often, transformations help align different functions into a familiar form.
In the problem at hand, simplifying the expression \( \tan \left( \frac{\pi}{4} - \frac{A}{2} \right) \) required angle transformation techniques. This was done by substituting a simplified expression into the angle of the tangent:

• The expression \( \frac{A}{2} = \frac{\pi}{4} - \tan^{-1} \sqrt{\tan \theta} \) transformed the original problem into a clearer structure for further simplification.

This change allows for a more straightforward calculation, as manipulating angle properties often leads to discovering more direct answers. Effectively using transformations can significantly reduce the complexity of dealing with more intricate expressions.

Tangent Function

The tangent function, expressed as \( \tan \), is one of the core trigonometric functions. It is a periodic function that repeats every \( \pi \, \text{radians} \), and this property makes it very useful in various areas of mathematics. The inverse tangent function, or \( \tan^{-1} \), essentially allows for computation of an angle when the tangent value is known.
In the exercise, the expression \( \tan \left( \tan^{-1} \sqrt{\tan \theta} \right) \) simplifies the whole problem considerably. Due to the property:

• \( \tan(\tan^{-1}(x)) = x \)

Thus, the sequence of function and its inverse leaves us with the original input, which is \( \sqrt{\tan \theta} \).
This property is fundamental when solving many trigonometric problems as it simplifies complex compositions into straightforward terms. Understanding and recognizing how tangent and its inverse function interact can make solving inverse trigonometric problems a seamless process.