Question

If \(\mathrm{x}>1,2 \tan ^{-1} \mathrm{x}\) is equal to (a) \(\tan ^{-1}\left(\frac{2 \mathrm{x}}{1-\mathrm{x}^{2}}\right)\) (b) \(\pi+\tan ^{-1}\left(\frac{2 x}{1-x^{2}}\right)\) (c) \(\tan ^{-1}\left(\frac{2 x}{1-x^{2}}\right)\) (d) \(\pi+\tan ^{-1} 2 \mathrm{x}\)

Short answer

Answer: (a) \(\tan^{-1}\left(\dfrac{2x}{1-x^2}\right)\)

Step-by-step solution

Recall the formula for tangent addition

Recall the formula for the tangent of the sum of two angles, which is given by: \[\tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A \cdot \tan B}\]

Express the given expression in terms of sum of angles

We can rewrite the given expression \(2 \tan ^{-1}x\) as a sum of angles, like this: \[\tan ^{-1}(x) + \tan ^{-1}(x)\]

Apply the formula to find the tangent

Let A = \(\tan ^{-1}(x)\) and B = \(\tan ^{-1}(x)\). Now apply the tangent addition formula: \[\tan(\tan ^{-1}(x) + \tan ^{-1}(x)) = \frac{\tan(\tan ^{-1}(x)) + \tan(\tan ^{-1}(x))}{1 - \tan(\tan ^{-1}(x)) \cdot \tan(\tan ^{-1}(x))}\]

Use identity \(\tan(\tan ^{-1}(x)) = x\)

Recall the identity \(\tan(\tan ^{-1}(x)) = x\), and substitute: \[\tan(2 \tan ^{-1}(x)) = \frac{x + x}{1 - x \cdot x}\]

Simplify the expression

Simplify this expression: \[\tan(2 \tan ^{-1}(x)) = \frac{2x}{1 - x^2}\]

Find the inverse tangent

Take the inverse tangent of both sides to get: \[2 \tan ^{-1}(x) = \tan^{-1}\left(\frac{2x}{1 - x^2}\right)\] From the given options, this expression matches option (a). Therefore, the correct answer is: (a) \(\tan ^{-1}\left(\frac{2 x}{1-x^{2}}\right)\)

Key concepts

Tangent Addition Formula

Understanding the tangent addition formula is crucial for solving many trigonometry problems, and it arises frequently in IIT-JEE preparation. This formula expresses the tangent of the sum of two angles as the sum of their tangents over one minus the product of their tangents. Essentially, it helps us combine two tangent values into one.

The formula is expressed as:
\[\begin{equation}\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \cdot \tan B}\end{equation}\]
For angles A and B, this formula allows the calculation of \(\tan(A + B)\) if \(\tan A\) and \(\tan B\) are known. This approach is extremely helpful when dealing with inverse trigonometric functions because they often require simplification using addition or subtraction of angles.

Remember, understanding and correctly applying the tangent addition formula is a powerful tool in simplifying complex trigonometric expressions, an essential skill in solving IIT-JEE trigonometry problems.

Trigonometry Problems

Trigonometry problems often form a significant part of mathematics syllabus for competitive exams like IIT-JEE. They challenge a student's problem-solving skills and understanding of trigonometric concepts. To tackle these problems effectively, one must be well-versed in various trigonometric identities, inverse functions, and their properties.

In the original exercise, the problem involves finding an equivalent expression for a function involving an inverse trigonometric function. As seen in the solution, it's necessary to recall and apply fundamental identities and formulas—like the tangent addition formula—to find a simplified form of the given function.

To succeed in trigonometry problems, focus on mastering the key formulas, understanding their derivations, and practicing their application in various scenarios. Moreover, you should work on visualizing the unit circle and trigonometric functions, as this can provide a deeper understanding and improve your ability to think critically about these problems.

IIT-JEE Mathematics Preparation

Preparing for the Mathematics section of the IIT-JEE requires a strategic and structured approach. The syllabus is vast, and the level of difficulty can be quite high. It is essential to have a thorough understanding of all topics, including algebra, calculus, and trigonometry. As we explored with inverse trigonometric functions and the tangent addition formula, in-depth knowledge and application skills are essential.

Preparation Tips:

• Start with the basics and ensure your foundational concepts are strong.
• Practice rigorously with a variety of problems to build speed and accuracy.
• Focus on understanding the logic behind formulas and theorems, rather than just memorizing them.
• Review and solve previous year IIT-JEE papers.
• Do not ignore the importance of timed practice tests to simulate exam conditions and improve time management.

Remember, mathematics is a subject that gets better with practice. Stay consistent, and gradually increase the level of difficulty in your practice sessions for a result-oriented preparation journey toward the IIT-JEE.