Problem 46 - A Complete Resource Book in Mathematics for JEE Main

Question

The value of \(x\) for which \(\sin \left(\cot ^{-1}(1+x)\right)=\cos \left(\tan ^{-1} x\right)\) is (A) \(\frac{1}{2}\) (B) 1 (C) 0 (D) \(-\frac{1}{2}\)

Step-by-Step Solution

Answer

The value of \(x\) is \(\frac{1}{2}\).

Step 1: Understanding the Inverse Trigonometric Functions

The problem involves inverse trigonometric functions \(\cot^{-1}\) and \(\tan^{-1}\). \(\cot^{-1}(1+x)\) returns an angle whose cotangent is \(1+x\), and \(\tan^{-1}(x)\) returns an angle whose tangent is \(x\). We need to simplify the expressions \(\sin\) and \(\cos\) to find a common value of \(x\).

Step 2: Convert Inverse Functions to Angles

Let \(\theta = \cot^{-1}(1+x)\) and \(\phi = \tan^{-1}(x)\). By definition, \(\cot(\theta) = 1 + x\) implies \(\tan(\theta) = \frac{1}{1+x}\), while \(\tan(\phi) = x\).

Step 3: Use the Identity \(\sin(\theta) = \cos(\phi)\)

Since \(\sin(\cot^{-1}(1+x)) = \cos(\tan^{-1}(x))\), we set \(\sin(\theta) = \cos(\phi)\). In trigonometric terms, this identity checks if \( \theta + \phi = \frac{\pi}{2}\) or \( \theta - \phi = 0\).

Step 4: Check the Angle Identity

This condition \(\theta + \phi = \frac{\pi}{2}\) implies that \(\tan(\theta) = \cot(\phi)\). Thus, if \( \tan(\theta) = \frac{1}{1+x} \) and \( \tan(\phi) = x\), solving \(\frac{1}{1+x} = \frac{1}{x}\) leads to the condition \(x = \frac{1}{2}\).

Step 5: Verify the Solution

Substitute \(x = \frac{1}{2}\) back into the trigonometric identities: \(\cot^{-1}(1+\frac{1}{2}) = \cot^{-1}(\frac{3}{2})\) and \(\tan^{-1}(\frac{1}{2})\). Verify that \(\sin(\cot^{-1}(\frac{3}{2})) = \cos(\tan^{-1}(\frac{1}{2}))\), confirming the condition holds.

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental tools in mathematics used to simplify expressions involving trigonometric functions like sine, cosine, tangent, and cotangent. These identities help establish relationships between different trigonometric functions, especially when dealing with inverse trigonometric functions. In this exercise, understanding identities is key to solving the equation \[\sin(\cot^{-1}(1+x)) = \cos(\tan^{-1}(x))\]By using identities, we can rewrite and transform complex expressions into simpler forms, making it easier to find solutions.

• Pythagorean Identities: These relate the square of sine and cosine, such as \( \sin^2(\theta) + \cos^2(\theta) = 1 \).
• Angle Sum and Difference Identities: These describe how to evaluate trigonometric expressions for sums and differences of angles, e.g. \( \sin(\alpha + \beta) = \sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta) \).

Using these identities correctly can reveal hidden relationships that facilitate solving equations or inequalities involving trigonometric functions.

Angle Identity

Angling identities involve angles formed by various trigonometric functions, which can simplify or directly solve equations. When dealing with the equation \[\sin(\cot^{-1}(1+x)) = \cos(\tan^{-1}(x))\]it represents a relationship between angles derived from inverse trigonometric functions. By setting \(\theta = \cot^{-1}(1+x)\) and \(\phi = \tan^{-1}(x)\), we explore angles whose properties can be unraveled using angle identities.

• Sum of Angles: The identity \( \theta + \phi = \frac{\pi}{2} \) implies that their corresponding trigonometric functions are complements.
• Difference of Angles: This can also help to verify if a solution holds true, validating the initial condition or equation we have.

Understanding angle identities thus aids in utilizing relationships between angles and trigonometric functions to reach the desired result effectively.

Cotangent Function

The cotangent function, denoted as \(\cot(\theta)\), is the reciprocal of the tangent function. For any angle \(\theta\), it is defined as \[\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{\cos(\theta)}{\sin(\theta)}\]This function is crucial in our problem as it helps us define \(\theta = \cot^{-1}(1+x)\), meaning we are dealing with an angle whose cotangent is \(1+x\).

• Inverse Cotangent: The inverse cotangent function \(\cot^{-1}(x)\) returns the angle whose cotangent is \(x\).
• Periodic Nature: Understanding the periodicity of the cotangent function aids in predicting behavior across different cycles or rotations.

The function assists in connecting angles derived from other trigonometric functions, allowing calculations to move forward.

Tangent Function

The tangent function, denoted as \(\tan(\theta)\), is the ratio of the sine to the cosine of an angle \(\theta\): \[\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\]This is especially significant in problems involving inverse trigonometric functions. Here, we use it by setting \(\phi = \tan^{-1}(x)\), which implies \[\tan(\phi) = x\]

• Inverse Tangent: The function \(\tan^{-1}(x)\) provides the angle whose tangent is \(x\).
• Applications: It is extensively used in geometry and calculus, helping to determine slopes and angles.

The tangent function and its inverse give critical insight into angle measurement and their comparison, essential for solving trigonometric equations.