Problem 36 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(-1

Step-by-Step Solution

Answer

Option C: \\(-\cot^{-1}\left(\frac{\sqrt{1-x^{2}}}{x}\right)\\)."

Step 1: Understanding the range of inverse tangent

The function \(\tan^{-1} x\) gives an angle whose tangent is \(-1 < x < 0\), which means it lies in the fourth quadrant of the unit circle or between \(-\frac{\pi}{2} < \theta < 0\). This is because \(\tan^{-1} x\) returns values in the range of \((-\frac{\pi}{2}, \frac{\pi}{2})\).

Step 2: Analyze the options

We need to match \(\tan^{-1} x\) with one of the given expressions. Look closely at the forms that involve trigonometric identities or transformations of the inverse tangent function.

Step 3: Evaluate each option

- **Option A**: \(\pi - \cos^{-1}(\sqrt{1-x^2})\) does not work since \(\cos^{-1}(y)\) output lies in \([0, \pi]\) and subtracting from \(\pi\) leads to invalid angles for \(\tan^{-1}(x)\, x < 0\).- **Option B**: \(\sin^{-1}\left(\frac{x}{\sqrt{1+x^2}}\right)\) doesn't represent \(\tan^{-1} x\).- **Option C**: \(-\cot^{-1}\left(\frac{\sqrt{1-x^2}}{x}\right)\) is a transformation that typically corresponds to \(\tan^{-1} x\).- **Option D**: \(\operatorname{cosec}^{-1} x\) is not defined for negative values of x like \(\tan^{-1} x\).

Step 4: Verify option C

The formula \(\tan^{-1}x = -\cot^{-1}(\frac{1}{x})\), and for given bounds, correctly reflects the range of inverse cotangent. Therefore, option C matches \(\tan^{-1} x\) perfectly.

Key Concepts

Inverse Tangent Function

The inverse tangent function, often denoted as \( an^{-1} x\), or arctan, is a crucial component in trigonometry, especially when dealing with angle calculation. Unlike some functions, the range of \( an^{-1} x\) is not the full set of angles on the unit circle. Instead, it specifically covers the interval \((-rac{\pi}{2}, \frac{\pi}{2})\).
This means the inverse tangent will produce an angle in the fourth or first quadrants, where tangent values are from negative infinity to positive infinity. In practical terms, it's useful when you need to find an angle whose tangent is a given number.
With \(-1 < x < 0\), the angle found using \( an^{-1} x\) will fall in the fourth quadrant. Here, the tangent values are negative, aligning with our given condition. Therefore, be mindful that the angle will be a negative value because it's below the x-axis.

Trigonometric Identities

Trigonometric identities are formulas that enable the manipulation and transformation of trigonometric expressions. These identities are like mathematical shortcuts that allow you to simplify complex trigonometric problems.
In this exercise, we observe transformations of \(\tan^{-1} x\) to expressions involving other inverse trigonometric functions. A notable identity used is the reciprocal relationship between tangent and cotangent: \(\tan^{-1}x = -\cot^{-1}igg(\frac{1}{x}\bigg)\). This identity is handy, especially when the range and quadrant considerations are crucial.
Seeing an option like \(-\cot^{-1}(\frac{\sqrt{1-x^2}}{x})\) reveals a manipulation that makes use of known identities for simplification. Understanding and employing these identities can be a huge advantage in solving trigonometric equations.

Quadrants of Unit Circle

The unit circle is a fundamental tool in trigonometry that helps visualize and understand angles and their corresponding trigonometric values. It is divided into four quadrants, each with specific characteristics.
Here's a quick view of the quadrants:

• **First Quadrant (Quadrant I):** All trigonometric functions are positive.
• **Second Quadrant (Quadrant II):** Sine is positive; cosine and tangent are negative.
• **Third Quadrant (Quadrant III):** Tangent is positive; sine and cosine are negative.
• **Fourth Quadrant (Quadrant IV):** Cosine is positive; sine and tangent are negative.

For \(-1 < x < 0\), \(\tan^{-1} x\) results in an angle in Quadrant IV. In this quadrant, the angle is negative, confirming it lies below the x-axis, aligning with the range of inverse functions discussed earlier.
This quadrant-based understanding allows us to predict the behavior and sign of various trigonometric functions and to correctly interpret results for angles calculated using inverse trigonometric functions.