Problem 53 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

\(2 \tan ^{-1}\left(\tan \frac{\theta}{2} \tan \frac{\phi}{2}\right)=\) (A) \(\cos ^{-1}\left(\frac{\cos \theta+\cos \phi}{1+\cos \theta \cos \phi}\right)\) (B) \(\cos ^{-1}\left(\frac{\cos \theta-\cos \phi}{1+\cos \theta \cos \phi}\right)\) (C) \(\cos ^{-1}\left(\frac{\cos \theta+\cos \phi}{1-\cos \theta \cos \phi}\right)\) (D) none of these

Step-by-Step Solution

Answer

Option (A).

Step 1: Understand the Problem Statement

We need to evaluate the expression \(2 \tan^{-1}\left(\tan \frac{\theta}{2} \tan \frac{\phi}{2}\right)\) and compare it to the given options to see which matches, if any.

Step 2: Recall a Useful Trigonometric Identity

The identity \( \tan^{-1}(x) + \tan^{-1}(y) = \tan^{-1}\left(\frac{x + y}{1 - xy}\right)\), when \(xy < 1\), might be helpful in evaluating expressions involving inverse tangent functions.

Step 3: Apply the Identity to Simplify the Expression

In this expression, consider setting \( x = \tan \frac{\theta}{2} \) and \( y = \tan \frac{\phi}{2} \). We seek a form of \( 2 \tan^{-1}(z) \), where \( z = x \cdot y \). Substitute \( z \) back: \( 2 \tan^{-1}(z) = \tan^{-1}\left(\frac{2z}{1 - z^2}\right)\).

Step 4: Simplify Right Hand Side and Connect to Options

To compare, reformulate \( \tan^{-1}\left(\frac{2z}{1 - z^2}\right)\) into a known cosine identity using trigonometric identities.

Step 5: Derive the Expression for Cosine Relation

Recall the cosine sum identity: \( \cos(A + B) = \cos A \cos B - \sin A \sin B \). The expression rewrites using \( \,\cos^{-1}x \, \) which relates to \( x = \frac{\cos \theta + \cos \phi}{1 + \cos \theta \cos \phi} \), connecting directly with option (A).

Step 6: Validate the Matching Option

Check each option using possible substitutions and confirm that \( \cos^{-1}\left(\frac{\cos \theta + \cos \phi}{1 + \cos \theta \cos \phi}\right)\) matches with the derived cosine relation, confirming it matches option (A).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variable. They are useful for simplifying and solving trigonometric expressions. One of the core identities used here is for inverse tangent. When dealing with expressions like \(2 \tan^{-1} (x)\), we utilize identities to simplify them.To handle inverse tangent, a handy identity is \( \tan^{-1}(x) + \tan^{-1}(y) = \tan^{-1} \left(\frac{x + y}{1 - xy} \right)\). This identity works when the product \( x \cdot y < 1 \).The steps often involve substituting values into this identity to express compound angles in terms of known trigonometric functions.

• Useful for simplifying expressions and equations.
• Aids in transformations between coordinate forms.

This foundational understanding allows for the conversion between different trigonometric forms, making it easier to identify equivalent expressions or solve equations.

Angle Sum and Difference Identities

Angle sum and difference identities help us find the trigonometric values of combined angles. They break down complex expressions into sums or differences of simpler trigonometric terms.Take the cosine sum identity for instance: \( \cos(A + B) = \cos A \cos B - \sin A \sin B \). This allows us to express the cosine of a sum of angles in terms of the product of sine and cosine of the individual angles.In practice, these identities are key when simplifying expressions involving multiple angles or when dealing with inverse functions.

• Facilitate solving trigonometric equations.
• Enable conversion of product identities to sum or difference identities.

These tools are powerful in trigonometry, especially when combined with inverse trigonometric functions to reach a solution that aligns with standard trigonometric forms.

Cosine Function

The cosine function is one of the fundamental trigonometric functions. It is especially useful for identifying relationships between angles and side lengths in right triangles.The inverse cosine function, or \(\cos^{-1}(x)\), returns an angle whose cosine value is \(x\). This is particularly useful when trying to match expressions to their equivalent cosines.When working with problems like the original exercise, recognizing that certain expressions can be rewritten using the cosine function is needed. For example, connecting inverse tangent results to the cosine via the identity: \( \cos^{-1}\left(\frac{\cos \theta + \cos \phi}{1 + \cos \theta \cos \phi}\right)\).

• Answers how angles relate in cosine form.
• Provides a way to reconstruct angle measurements from numerical cosines.

Understanding the role of cosine and its inverse is crucial for solving equations where angles are expressed in terms of their cosine values, especially when confirming solutions like in the original exercise.