Problem 12 - Calculus 1 Course in Mathematics for the IIT JEE and Other Engineering Exams

Question

If \(f(\theta)=\tan \theta\), then \(\frac{f(\theta)-f(\phi)}{1+f(\theta) f(\phi)}\) is equal to (a) \(f(\theta-\phi)\) (b) \(f(\phi-\theta)\) (c) \(f(\theta+\phi)\) (d) None of these

Step-by-Step Solution

Answer

(a) \( f(\theta - \phi) \)

Step 1: Understanding the Problem

We need to find the expression for \( \frac{f(\theta)-f(\phi)}{1+f(\theta)f(\phi)} \) where \( f(\theta) = \tan \theta \). We know that tangent addition formulas can be used in trigonometry to simplify such expressions.

Step 2: Applying Trigonometric Identity

Recall the identity for tangent of the sum of two angles: \( \tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b} \). We need an expression that resembles this identity but in the current expression, we should recognize the form of the difference identity: \( \tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b} \) similar to given expression.

Step 3: Match Given Expression with Identity

Compare our given expression \( \frac{f(\theta)-f(\phi)}{1+f(\theta)f(\phi)} = \frac{\tan \theta - \tan \phi}{1 + \tan \theta \tan \phi} \) to the identity \( \tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b} \). It's a perfect match with the difference of tangent identity.

Step 4: Conclusion

Since our expression matches \( \tan(\theta - \phi) \), the expression \( \frac{f(\theta)-f(\phi)}{1+f(\theta)f(\phi)} \) simplifies to \( f(\theta - \phi) \). Thus, the answer is option (a).

Key Concepts

Tangent Function

The tangent function is one of the fundamental trigonometric functions in mathematics. It is often abbreviated as \( \tan \theta \) when referring to the tangent of an angle \( \theta \). The tangent function relates the angles in a right triangle to the ratio of two sides. Specifically, for an angle \( \theta \) in a right triangle:

• The tangent is the ratio of the length of the opposite side to the length of the adjacent side.
• Symbolically, this is expressed as \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \).
• Tangent values are often tabulated alongside sine and cosine for quick reference by students and practitioners.

Understanding the behavior of the tangent function is essential in solving trigonometry problems. Unlike sine and cosine, which have values between -1 and 1, tangent can take any real number value. This makes it unique and sometimes challenging to work with, particularly near its undefined points when \( \theta \) is an odd multiple of \( 90^\circ \) (or \( \frac{\pi}{2} \) radians). Therefore, special attention is needed when using tangent in various trigonometric identities.

Angle Difference Identity

In trigonometry, angle identities help simplify expressions involving trigonometric functions of multiple angles. One such important identity is the angle difference identity for tangent. It describes how the tangent of a difference of two angles can be expressed relative to the tangents of the two individual angles:

• The identity is given by \( \tan(\theta - \phi) = \frac{\tan \theta - \tan \phi}{1 + \tan \theta \tan \phi} \).
• This equation is similar to the tangent addition formula, where the negative sign in the numerator and denominator distinguishes it from \( \tan(\theta + \phi) = \frac{\tan \theta + \tan \phi}{1 - \tan \theta \tan \phi} \).

By utilizing the angle difference identity, complex expressions can be simplified, like the one initially presented in the exercise. Such identities are fundamental when dealing with complex angles, especially in calculus and advanced trigonometry. Understanding these enables us to simplify and solve trigonometric problems efficiently.

Trigonometry Problem Solving

Trigonometry problem solving often involves reducing complex expressions using well-known identities. In the specific exercise we are looking at, recognizing and applying the correct trigonometric identities was key to solving the problem correctly.

• The problem required us to identify a form of the expression, \( \frac{f(\theta)-f(\phi)}{1+f(\theta)f(\phi)} \), and compare it with familiar trigonometric identities.
• The angle difference identities for tangent, such as \( \tan(\theta - \phi) \), were crucial in simplifying the expression.
• Understanding when and how to apply these identities allowed us to correctly identify option (a) as the solution.

The importance of recognizing patterns and matching them to known identities cannot be overstated in problem-solving scenarios. Such skills form the foundation for solving a wide range of mathematical problems efficiently and accurately, leading to greater success in understanding and applying trigonometric concepts.