Problem 31 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The value of \(\cot \left(\operatorname{cosec}^{-1} \frac{5}{3}+\tan ^{-1} \frac{2}{3}\right)\) is (A) \(6 / 17\) (B) \(3 / 17\) (C) \(4 / 17\) (D) \(5 / 17\)

Step-by-Step Solution

Answer

(A) \(6/17\)

Step 1: Understand the problem

We need to find the value of \( \cot \left( \csc^{-1} \frac{5}{3} + \tan^{-1} \frac{2}{3} \right) \). This involves inverse trigonometric functions, so we should familiarize ourselves with their definitions and properties.

Step 2: Simplify \( \csc^{-1} \left( \frac{5}{3} \right) \)

Let \( \theta = \csc^{-1} \left( \frac{5}{3} \right) \). Then, \( \csc \theta = \frac{5}{3} \). We know \( \csc \theta = \frac{1}{\sin \theta} \), so \( \sin \theta = \frac{3}{5} \). Using Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \), we find \( \cos \theta = \sqrt{1 - \left( \frac{3}{5} \right)^2} = \frac{4}{5} \).

Step 3: Simplify \( \tan^{-1} \left( \frac{2}{3} \right) \)

Let \( \phi = \tan^{-1} \left( \frac{2}{3} \right) \). This means \( \tan \phi = \frac{2}{3} \). We can represent a right triangle where opposite side = 2, adjacent side = 3, so the hypotenuse is \( \sqrt{2^2 + 3^2} = \sqrt{13} \). Thus, \( \sin \phi = \frac{2}{\sqrt{13}} \) and \( \cos \phi = \frac{3}{\sqrt{13}} \).

Step 4: Identify the components of the main expression

We need to evaluate \( \cot \left( \theta + \phi \right) \). To do this, use the identity \( \cot(\alpha + \beta) = \frac{\cot \alpha \cot \beta - 1}{\cot \alpha + \cot \beta} \). First, calculate \( \cot \theta = \frac{\cos \theta}{\sin \theta} = \frac{4}{3} \) and \( \cot \phi = \frac{3}{2} \).

Step 5: Calculate \( \cot(\theta + \phi) \)

Using \( \cot(\theta + \phi) = \frac{\cot \theta \cot \phi - 1}{\cot \theta + \cot \phi} \), plug in \( \cot \theta = \frac{4}{3} \) and \( \cot \phi = \frac{3}{2} \). This gives: \[ \cot(\theta + \phi) = \frac{\left( \frac{4}{3} \right) \left( \frac{3}{2} \right) - 1}{\frac{4}{3} + \frac{3}{2}} = \frac{2 - 1}{\frac{8}{6} + \frac{9}{6}}. \]

Step 6: Simplify the expression

Simplifying the calculations from the previous step: \[ \cot(\theta + \phi) = \frac{1}{\frac{17}{6}} = \frac{6}{17}. \] Therefore, the value of \( \cot \left( \theta + \phi \right) \) is \( \frac{6}{17} \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are the reverse operations of standard trigonometric functions, providing angles when the function values are known. They include functions like \( \sin^{-1} \), \( \cos^{-1} \), and \( \tan^{-1} \), where each yields an angle for a given trigonometric ratio.

• For instance, \( \csc^{-1} \left( \frac{5}{3} \right) \) means we are looking for an angle \( \theta \) such that \( \csc \theta = \frac{5}{3} \).
• Similarly, \( \tan^{-1} \left( \frac{2}{3} \right) \) indicates an angle \( \phi \) where \( \tan \phi = \frac{2}{3} \).

Inverse functions are essential because they enable the calculation of angles based on given trigonometric values, crucial for solving problems in geometry and calculus. Remember, their outputs are specific ranges to ensure they remain as functions.
Expressing problems through inverse functions helps in breaking down complex trigonometric equations into simpler, solvable parts. It's important to understand their practical ranges and constraints for accurate computation and interpretation.

Pythagorean Identity

The Pythagorean Identity is one of the fundamental principles in trigonometry, stating that for any angle \( \theta \), the equation \( \sin^2 \theta + \cos^2 \theta = 1 \) is always true. This identity helps to relate the sine and cosine functions directly to each other.

• In the provided exercise, we use this identity to find \( \cos \theta \) given that \( \sin \theta = \frac{3}{5} \).
• Using the identity, \( \cos \theta = \sqrt{1 - \left( \frac{3}{5} \right)^2} = \frac{4}{5} \).

This powerful relation is instrumental in transforming and simplifying complex trigonometric equations, making it one of the most frequently used identities in trigonometry. Understanding how to apply the Pythagorean Identity efficiently is crucial for finding unknown trigonometric values quickly.

Cotangent Addition Formula

The Cotangent Addition Formula helps in calculating the cotangent of a sum of two angles. It is expressed as:\[\cot(\alpha + \beta) = \frac{\cot \alpha \cot \beta - 1}{\cot \alpha + \cot \beta}.\]This formula is especially useful when dealing with expressions involving multiple angle additions.

• In our exercise, we applied this formula to compute \( \cot(\theta + \phi) \), knowing \( \cot \theta = \frac{4}{3} \) and \( \cot \phi = \frac{3}{2} \).
• Substituting these values gives \( \cot(\theta + \phi) = \frac{\left( \frac{4}{3} \right) \left( \frac{3}{2} \right) - 1}{\frac{4}{3} + \frac{3}{2}} \), and further simplification leads to \( \frac{6}{17} \).

Understanding this formula is crucial for dealing with scenarios where direct computation of an angle summation through standard trigonometric functions feels cumbersome. It paved the way for an elegant and straightforward computation in our example.