Problem 12 - A Complete Resource Book in Mathematics for JEE Main

Question

The solution of \(\sin ^{8} x+\cos ^{8} x=\frac{17}{32}\) is (A) \(\frac{n \pi}{2} \pm \frac{\pi}{8}\) (B) \(n \pi \pm \frac{\pi}{4}\) (C) \(n \pi \pm \frac{\pi}{8}\) (D) no solution

Step-by-Step Solution

Answer

The solution is (A) \( \frac{n \pi}{2} \pm \frac{\pi}{8} \).

Step 1: Understanding the Equation

We are given the equation \( \sin^8 x + \cos^8 x = \frac{17}{32} \). This equation involves trigonometric functions raised to a power, and we are looking to find the values of \( x \) that satisfy this equation.

Step 2: Use Trigonometric Identity

Utilize the identity \( \sin^2 x + \cos^2 x = 1 \). First, express \( \sin^8 x + \cos^8 x \) in terms of simpler powers by considering it in a squared form: \[(\sin^4 x + \cos^4 x)^2 = \sin^8 x + \cos^8 x + 2 \sin^4 x \cos^4 x.\] Thus, we have:\[\sin^8 x + \cos^8 x = (\sin^4 x + \cos^4 x)^2 - 2\sin^4 x \cos^4 x.\]

Step 3: Simplify Using Power Reduction Identities

Using the identity \( \sin^4 x + \cos^4 x = \frac{(\,\sin^2 x + \cos^2 x)^2 - 2\sin^2 x \cos^2 x}{2} = \frac{1 - 2\sin^2 x \cos^2 x}{2}\), we substitute this into our expression:\[\sin^8 x + \cos^8 x = \left( \frac{1 - 2\sin^2 x \cos^2 x}{2} \right)^2 - 2\sin^4 x \cos^4 x.\]

Step 4: Substitute and Solve

Express \( \sin^2 x \cos^2 x \) using the identity \( \sin^2 x \cos^2 x = \frac{1}{4} \sin^2 2x \). Therefore, \[\sin^8 x + \cos^8 x = \left( \frac{1 - \sin^2 2x/2}{2} \right)^2 - \frac{1}{8} \sin^4 2x.\]Comparing this with \( \frac{17}{32} \) and solving for \( \sin^2 2x \), we find that \( \sin^2 2x = \frac{1}{2} \).

Step 5: Find General Solution for Trigonometric Equation

The general solution for \( \sin^2 2x = \frac{1}{2} \) implies \( \sin 2x = \pm \frac{\sqrt{2}}{2} \). Thus, \( 2x = n\pi + (-1)^n \frac{\pi}{4} \). Solving for \( x \), we find:\[x = \frac{n \pi}{2} + \frac{(-1)^n \pi}{8}.\]

Step 6: Matching Options

The solution format derived \( x = \frac{n \pi}{2} \pm \frac{\pi}{8} \) matches with option (A). Therefore, the correct answer is (A).

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental tools used to simplify and solve trigonometric equations. They express relationships between different trigonometric functions, helping us break down complex expressions into simpler forms. One of the most essential trigonometric identities is:

• \( \sin^2 x + \cos^2 x = 1 \)

This identity is commonly used because it relates the sine and cosine functions, which are the building blocks of many trigonometric equations. In our exercise, we utilized a variation of this identity to express higher powers of sine and cosine in more manageable terms.
By leveraging such identities, we can transform complex expressions like \( \sin^8 x + \cos^8 x \) into simpler, more workable forms, which are easier to address using algebraic methods. Recognizing and applying the right trigonometric identities is crucial in efficiently tackling these equations.

Power Reduction

Power reduction is a technique used to simplify the power of trigonometric functions by expressing them in terms of functions with lower powers. This often involves using known identities to reduce the complexity of the expressions.
In the given problem, we used the identity:

• \( \sin^4 x + \cos^4 x = \frac{(\sin^2 x + \cos^2 x)^2 - 2\sin^2 x \cos^2 x}{2} \)

Combining this with the identity for \( \sin^2 x \cos^2 x \), we could express higher powers in a reduced form. This allows us to analytically manipulate expressions that might seem daunting at first glance.
The ability to reduce powers efficiently makes solving complex trigonometric equations much more manageable and is an invaluable technique when dealing with equations involving higher powers.

General Solution

When dealing with trigonometric equations, determining the general solution is a key step. A general solution describes all possible values that satisfy the equation, not just the principal values. This approach is vital in capturing the periodic nature of trigonometric functions.
In the exercise, once the equation \( \sin^2 2x = \frac{1}{2} \) was solved, we derived the general solution form:

• \( x = \frac{n \pi}{2} + \frac{(-1)^n \pi}{8} \)

This formulation reflects the periodic pattern of the sine function and accounts for all instances when the equation holds true. By finding the general solution, we ensure that our solution set is complete and accounts for all cycles of the trigonometric function.
Understanding and finding the general solution is essential for fully capturing the solutions of trigonometric equations, particularly those that span multiple cycles or angles.