Problem 20 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The value of \(\tan ^{-1}\left[\frac{\sqrt{1+x^{2}}+\sqrt{1-x^{2}}}{\sqrt{1+x^{2}}-\sqrt{1-x^{2}}}\right],|x|<\frac{1}{2}, x \neq 0\), is equal to [Online April 8, 2017] (a) \(\frac{\pi}{4}+\frac{1}{2} \cos ^{-1} \mathrm{x}^{2}\) (b) \(\frac{\pi}{4}+\cos ^{-1} x^{2}\) (c) \(\frac{\pi}{4}-\frac{1}{2} \cos ^{-1} \mathrm{x}^{2}\) (d) \(\frac{\pi}{4}-\cos ^{-1} x^{2}\)

Step-by-Step Solution

Answer

The answer is option (a): \( \frac{\pi}{4} + \frac{1}{2} \cos^{-1} x^{2} \).

Step 1: Identify the Form of Expression

We need to simplify the expression \( \frac{\sqrt{1+x^{2}}+\sqrt{1-x^{2}}}{\sqrt{1+x^{2}}-\sqrt{1-x^{2}}} \). Begin by examining components: numerators and denominators using trigonometric identities and simplification.

Step 2: Simplifying Using Identical Forms

Recognize that the expression resembles a difference of two squares. Rewrite as:\[ \frac{(\sqrt{1+x^2})^2 - (-\sqrt{1-x^2})^2^{1/2}}{\sqrt{1+x^2}-\sqrt{1-x^2}} = \frac{(1+x^2) - (1-x^2)}{(\sqrt{1+x^2} - \sqrt{1-x^2})}\times (\sqrt{1+x^2} + \sqrt{1-x^2}); \]which simplifies to:\[ \frac{2x^2}{\sqrt{1+x^2}-\sqrt{1-x^2}}. \]

Step 3: Applying Trigonometric Transformations

Express the numerator in a trigonometric form. Set:\[ \tan A = \frac{\sqrt{1-x^{2}}}{\sqrt{1+x^{2}}}, \]and therefore,\[ A = \tan^{-1}\left(\frac{\sqrt{1-x^{2}}}{\sqrt{1+x^{2}}}\right). \]

Step 4: Use Sum and Difference Formulas

Utilize angle addition formulas:\[ \tan(45^\circ + A) = 1+ A, \]Using identities:\[ \frac{c+b}{c-b} = \tan(\frac{\pi}{4} + \cos^{-1} x), \]This matches with problems with setup for inverse tangent equality.

Step 5: Concluding the Calculation

Recognize from prior steps that this results in the Arctangent identity solving to:\(\tan^{-1}\left[ \frac{\sqrt{1+x^{2}}+\sqrt{1-x^{2}}}{\sqrt{1+x^{2}}-\sqrt{1-x^{2}}} \right] = \frac{\pi}{4} + \frac{1}{2} \cos^{-1}(x^{2}). \)

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for every value of the involved variables, where both sides of the equation are defined. These identities are fundamental in simplifying or solving trigonometric expressions and equations.

There are several important trigonometric identities, but some of the most useful include:

• Pythagorean Identity: \( \sin^2 \theta + \cos^2 \theta = 1 \).
• Reciprocal Identities: such as \( \csc \theta = \frac{1}{\sin \theta} \).
• Quotient Identities: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).

In the original exercise, recognizing that the expression resembles a form like a difference of squares is crucial. This allows us to apply these identities to simplify the original trigonometric expression, eventually leading to understanding the angle transformations involved. Knowing these identities can greatly help in managing and breaking down complex trigonometric equations into solvable components.

Angle Addition Formulas

Angle addition formulas in trigonometry are equations that provide the trigonometric functions for the sum or difference of two angles. These formulas are quite helpful in simplifying complicated trigonometric expressions and performing transformations.

For instance, the angle addition formula for tangent is:

• \( \tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B} \)

This formula allows you to compute the tangent of an angle made up of two different angles, which was used in the exercise to evaluate the expression:\( \tan(45^\circ + A) = 1 + A \). Here, we equate this to the provided expression to make transformations and simplify.

Studying these formulas can enhance your ability to manipulate trigonometric expressions and discover the identities within more complex algebraic structures.

Simplifying Expressions

Simplifying expressions, especially in trigonometry, involves using known identities and algebraic manipulation to rewrite expressions in a more manageable or insightful form. This process can reveal new relationships or make it easier to evaluate an expression for specific values.

In the provided exercise, the initial complex expression \( \frac{\sqrt{1+x^{2}}+\sqrt{1-x^{2}}}{\sqrt{1+x^{2}}-\sqrt{1-x^{2}}} \) was rewritten and reduced through steps like:

• Applying known identities.
• Recognizing patterns such as the difference of squares.
• Breaking down the components into simpler terms.

As we simplified, it emerged that the trigonometric identity matched the arctangent, \( \tan^{-1} \), leading to a correct angle transformation that eventually resulted in: \( \frac{\pi}{4} + \frac{1}{2} \cos^{-1}(x^{2}) \).

Simplifying such expressions helps not only in finding solutions, but also in developing a deeper understanding of how different mathematical concepts interplay.