Problem 50 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The value of \(\frac{1-\tan ^{2} 15^{\circ}}{1+\tan ^{2} 15^{\circ}}\) is: \(\quad\) (A) 1 (B) \(\sqrt{3}\) (C) \(\frac{\sqrt{3}}{2}\) (D) 2

Step-by-Step Solution

Answer

The value is \( \frac{\sqrt{3}}{2} \) (Option C).

Step 1: Understand the Identity

First, we need to recall the identity for tan in terms of sum and difference formulas. The identity \( 1 - \tan^2(\theta) = \frac{\cos(2\theta)}{\cos^2(\theta)} \) and \( 1 + \tan^2(\theta) = \frac{1}{\cos^2(\theta)} \) can help us simplify the expression.

Step 2: Use Double Angle Identity for Cosine

By the identity, we know \( \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) \), which is also \( 1 - 2\sin^2(\theta) = 2\cos^2(\theta) - 1 \). For \( \theta = 15^\circ \), \( 2\theta = 30^\circ \), so \( \cos(30^\circ) = \frac{\sqrt{3}}{2} \).

Step 3: Simplify the Expression

Substituting the double angle result: \( 1 - \tan^2(15^\circ) = \cos(30^\circ) \) and using \( 1 + \tan^2(15^\circ) = 1 / \cos^2(15^\circ) \). The expression becomes \( \frac{\cos(30^\circ)}{1} = \cos(30^\circ) \).

Step 4: Evaluate the Expression

Now we evaluate \( \cos(30^\circ) = \frac{\sqrt{3}}{2} \). So the value of the original expression \( \frac{1 - \tan^2(15^\circ)}{1 + \tan^2(15^\circ)} \) is \( \frac{\sqrt{3}}{2} \).

Key Concepts

Double Angle Formulas

Double angle formulas are vital tools in trigonometry that help simplify expressions. They relate functions of double angles (such as \(2\theta\)) to functions of a single angle \(\theta\). For trigonometric functions like sine, cosine, and tangent, knowing these formulas helps us recalibrate and evaluate expressions more easily. The double angle formula for cosine, which is heavily used in our exercise, states:

• \(\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)\)
• Alternatively, \(\cos(2\theta) = 1 - 2\sin^2(\theta)\)
• Or \(\cos(2\theta) = 2\cos^2(\theta) - 1\)

These variations allow different approaches depending on the given conditions, and in this exercise, they provide concrete paths to simplifying trigonometric problems. For \(\theta = 15^\circ\), \(\cos(30^\circ) = \frac{\sqrt{3}}{2}\) is determined using the double angle formula.

Trigonometric Simplification

Trigonometric simplification involves transforming complex expressions into simpler forms for ease of evaluation. This often requires employing various identities and formulas, as seen in the exercise where we focused on simplifying \(\frac{1 - \tan^2(15^\circ)}{1 + \tan^2(15^\circ)}\). Here are some key steps in simplification:

• Recognize the identities that apply, such as \(1 - \tan^2(\theta)\) and \(1 + \tan^2(\theta)\) in this scenario.
• Substitute known values: \(1 - \tan^2(\theta)\) can be rephrased using the cosine double angle identity.
• Gradually reduce the expression: The ultimate goal is to arrive at a straightforward expression such as \(\cos(30^\circ)\).

In essence, simplifying requires identifying relationships and reducing terms to known trigonometric values or other well-defined expressions.

Angle Transformation

Angle transformation refers to the process of changing the form of angles in order to aid calculations and simplifications. This can include doubling angles, as we did with \(15^\circ\) to get \(30^\circ\). Understanding how angles interrelate is crucial to solving trigonometric problems. In particular, angle transformation can involve:

• Using sum or difference identities to handle angles like \(\theta = 15^\circ\).
• Employing double angle or half-angle formulas for known values (e.g., turning \(15^\circ\) into \(30^\circ\)).

Transforming angles utilizes formulas to convert one form into another, ensuring the solution path aligns with known trigonometric identities for maximum efficiency.