Problem 198 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \tan ^{-1} x+2 \cot ^{-1} x=\frac{2}{3} \pi $$

Step-by-Step Solution

Answer

The solution to the equation is approximately \(x ≈ 3.7321\).

Step 1: Identify and Use Inverse Trig Identity

Recognize that \(\tan^{-1}(x) + \cot^{-1}(x) = \frac{\pi}{2}\) for \(x ≠ 0\). So substitute this identity into the equation: \(\frac{\pi}{2} + 2\cot^{-1}(x) = \frac{2\pi}{3}\).

Step 2: Isolate \(\cot^{-1}(x)\) term

Subtract \(\frac{\pi}{2}\) from both sides of the equation: \(2\cot^{-1}(x) = \frac{2\pi}{3} - \frac{\pi}{2} = \frac{\pi}{6}\). Then divide both sides by 2 to solve for \(\cot^{-1}(x)\) : \(\cot^{-1}(x) = \frac{\pi}{12}\). This means that \(\cot(x) = \frac{1}{\tan(x)} = \tan\left(\frac{\pi}{12}\right)\).

Step 3: Find the Value of \(x\)

Since \(\cot(x) = \tan\left(\frac{\pi}{12}\right)\), we therefore have \(x = \frac{1}{\tan\left(\frac{\pi}{12}\right)}\). We know \(\tan\left(\frac{\pi}{12}\right)\) is approximately 0.26794. So \(x ≈ \frac{1}{0.26794} ≈ 3.7321.\)

Key Concepts

Inverse Trig Identity

In trigonometry, inverse functions play a crucial role when reversing the process of a trig function to find an angle. An important identity involving the inverse trigonometric functions is that the sum of the inverse tangent and inverse cotangent of the same variable equals \( \frac{\pi}{2} \). More formally:

• \( \tan^{-1}(x) + \cot^{-1}(x) = \frac{\pi}{2} \) for \( x eq 0 \)

This identity can be quite handy when solving equations that involve both \( \tan^{-1}(x) \) and \( \cot^{-1}(x) \).
In particular, understanding and using these identities correctly allows for simplification of complex trigonometric equations, as demonstrated in the provided exercise solution.

Cotangent Function

The cotangent function, or \( \cot(x) \), is the reciprocal of the tangent function.
Functionally, it is defined as:

• \( \cot(x) = \frac{1}{\tan(x)} \)
• \( \cot(x) = \frac{\cos(x)}{\sin(x)} \)

The inverse cotangent, \( \cot^{-1}(x) \), is used when finding angles whose cotangent is \( x \).
This function often appears in geometric and trigonometric solutions where answers are derived from angles, especially when the tangent function is involved.

Tangent Function

The tangent function, represented by \( \tan(x) \), is a fundamental trigonometric function.
It is defined as the ratio of the sine and cosine of the angle:

• \( \tan(x) = \frac{\sin(x)}{\cos(x)} \)

The inverse tangent, \( \tan^{-1}(x) \), is the angle whose tangent gives the number \( x \).
This is especially useful when determining unknown angles from known values.
In this context, \( \tan^{-1} \) helps to find angles for a given tangent value, fitting perfectly into solving equations and applying inverse identities in trigonometric contexts.