Problem 5 - A Complete Resource Book in Mathematics for JEE Main

Question

The value of \(\sin \left(4 \tan ^{-1} \frac{1}{3}\right)-\cos \left(2 \tan ^{-1} \frac{1}{7}\right)\) is (A) \(\frac{4}{7}\) (B) 0 (C) \(\frac{7}{8}\) (D) none of these

Step-by-Step Solution

Answer

The value is \( \frac{1}{5} \), so the answer is (D) none of these.

Step 1: Substitute the Inverse Tangent Values

First, set \( x = \tan^{-1} \left( \frac{1}{3} \right) \) and \( y = \tan^{-1} \left( \frac{1}{7} \right) \). The problem reduces to calculating \( \sin(4x) - \cos(2y) \).

Step 2: Use the Angle Double Formulas

For \( \sin(4x) \), use the double angle formula twice. First, \( \sin(2x) = 2 \sin(x)\cos(x) \). Therefore, \( \sin(4x) = 2 \sin(2x)\cos(2x) = 4 \sin(x)\cos(x)(1 - 2 \sin^2(x)) \). For \( \cos(2y) \), use \( \cos(2y) = 1 - 2\sin^2(y) \).

Step 3: Calculate \( \sin(x) \) and \( \cos(x) \)

Since \( \tan(x) = \frac{1}{3} \), the corresponding right triangle has an opposite side of 1 and an adjacent side of 3, with hypotenuse \( \sqrt{10} \). Therefore, \( \sin(x) = \frac{1}{\sqrt{10}} \) and \( \cos(x) = \frac{3}{\sqrt{10}} \).

Step 4: Evaluate \( \sin(4x) \)

Using \( \sin(4x) = 4 \sin(x)\cos(x)(1 - 2 \sin^2(x)) \), substitute \( \sin(x) = \frac{1}{\sqrt{10}} \) and \( \cos(x) = \frac{3}{\sqrt{10}} \). Simplify to find \( \sin(4x) = \frac{4}{5} \).

Step 5: Calculate \( \sin(y) \) and \( \cos(y) \)

For \( \tan(y) = \frac{1}{7} \), the triangle has an opposite side of 1, an adjacent side of 7, and hypotenuse \( \sqrt{50} \). Thus, \( \sin(y) = \frac{1}{\sqrt{50}} \) and \( \cos(y) = \frac{7}{\sqrt{50}} \).

Step 6: Evaluate \( \cos(2y) \)

Use \( \cos(2y) = 1 - 2\sin^2(y) \) with \( \sin(y) = \frac{1}{\sqrt{50}} \). Simplifying gives \( \cos(2y) = \frac{3}{5} \).

Step 7: Subtract the Values

The expression simplifies to \( \sin(4x) - \cos(2y) = \frac{4}{5} - \frac{3}{5} = \frac{1}{5} \).

Step 8: Verify the Answer with Given Options

The value \( \frac{1}{5} \) does not match any of the provided options (\( \frac{4}{7} \), 0, \( \frac{7}{8} \), none of these). Therefore, the correct answer is \( \text{(D) none of these} \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions allow us to work backwards from known ratios to find angles. They are essential when a problem specifies a ratio, like \( \tan^{-1}(\frac{1}{3}) \), and asks for the corresponding angle. In this exercise, using the inverse tangent, or arctan, helps define angles based on given ratios.

Here's how they work:

• Arctan takes a ratio and returns an angle whose tangent is that ratio.
• For \( \tan^{-1}(\frac{1}{3}) \), we're finding an angle \( x \) such that \( \tan(x) = \frac{1}{3} \).
• Similarly, \( \tan^{-1}(\frac{1}{7}) \) provides an angle \( y \) such that \( \tan(y) = \frac{1}{7} \).

Recognizing and working with these inverse functions is crucial, particularly in transforming expressions or simplifying trigonometric forms.

Trigonometric Identities

Trigonometric identities are like formulas and rules that can simplify complex expressions or prove statements about angles. They are indispensable for transforming trigonometric expressions into more manageable forms.

In this problem, some key identities used are:

• The double-angle formula for sine: \( \sin(2x) = 2 \sin(x)\cos(x) \).
• The double-angle formula for cosine: \( \cos(2x) = 1 - 2\sin^2(x) \).

These identities allow us to rewrite multiples of angle functions in terms of basic sine and cosine functions. Using them:

• We can deduce \( \sin(4x) \) by applying the double-angle formula twice: \( \sin(4x) = 2 \sin(2x)\cos(2x) \).
• For \( \cos(2y) \), it becomes \( 1 - 2\sin^2(y) \) for simpler computations.

Understanding and manipulating such identities help solve problems efficiently and are foundational in trigonometry.

Angle Transformation

Angle transformation involves applying trigonometric formulas to change how angles are expressed or calculated in a problem. It involves recognizing how different forms can make calculations simpler.

In the exercise at hand, transformations like:\[ \sin(4x) = 4 \sin(x) \cos(x) (1 - 2\sin^2(x)) \]allow detailed expressions to become recognizable and easier to handle. Similarly, \[ \cos(2y) = 1 - 2\sin^2(y) \]simplifies the cosine into a function of sine, aiding easier subtraction. Here’s why angle transformations are pivotal:

• They help break down each component of complex angle expressions for straightforward computation.
• Such transformations reveal relationships and dependencies between functions.
• They support calculations that lead directly to the problem's solution.

Mastering angle transformations enables tackling complex trigonometric problems with ease and insight.