Problem 75 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(f_{k}(x)=\frac{1}{k}\left(\sin ^{k} x+\cos ^{4} x\right)\) where \(x \in R\) and \(k \geq 1\) then, the value of \(f_{4}(x)-f_{6}(x)\) equals (A) \(\frac{1}{6}\) (B) \(\frac{1}{3}\) (C) \(\frac{1}{4}\) (D) \(\frac{1}{12}\)

Step-by-Step Solution

Answer

The value of \(f_4(x) - f_6(x)\) is \(\frac{1}{12}\) (option D).

Step 1: Write down the given functions

We start by writing the expression for \(f_4(x)\) and \(f_6(x)\) using the given formula:\[f_4(x) = \frac{1}{4}(\sin^4 x + \cos^4 x)\]\[f_6(x) = \frac{1}{6}(\sin^6 x + \cos^4 x)\]

Step 2: Expand and simplify \(\sin^4 x\) and \(\sin^6 x\)

Note that simplifying \(\sin^4 x\) and \(\sin^6 x\) helps identify similarities and differences:\[\sin^4 x = (\sin^2 x)^2 = (1 - \cos^2 x)^2 = 1 - 2\cos^2 x + \cos^4 x\]\[\sin^6 x = (\sin^2 x)^3 = (1 - \cos^2 x)^3 = 1 - 3\cos^2 x + 3\cos^4 x - \cos^6 x\]

Step 3: Substitute \(\sin^4 x\) back into \(f_4(x)\)

We substitute the expression obtained from \(\sin^4 x\) into \(f_4(x)\):\[f_4(x) = \frac{1}{4}((1 - 2\cos^2 x + \cos^4 x) + \cos^4 x)\]Simplify to get:\[f_4(x) = \frac{1}{4}(1 - 2\cos^2 x + 2\cos^4 x)\]

Step 4: Substitute \(\sin^6 x\) back into \(f_6(x)\)

We substitute the expression obtained from \(\sin^6 x\) into \(f_6(x)\):\[f_6(x) = \frac{1}{6}(1 - 3\cos^2 x + 3\cos^4 x - \cos^6 x + \cos^4 x)\]Simplify to get:\[f_6(x) = \frac{1}{6}(1 - 3\cos^2 x + 4\cos^4 x - \cos^6 x)\]

Step 5: Calculate \(f_4(x) - f_6(x)\)

Subtract \(f_6(x)\) from \(f_4(x)\):\[f_4(x) - f_6(x) = \frac{1}{4}(1 - 2\cos^2 x + 2\cos^4 x) - \frac{1}{6}(1 - 3\cos^2 x + 4\cos^4 x - \cos^6 x)\]Combine and simplify terms:\[f_4(x) - f_6(x) = \frac{3(1 - 2\cos^2 x + 2\cos^4 x) - 2(1 - 3\cos^2 x + 4\cos^4 x - \cos^6 x)}{12}\]Continue simplifying:\[= \frac{3 - 6\cos^2 x + 6\cos^4 x - 2 + 6\cos^2 x - 8\cos^4 x + 2\cos^6 x}{12}\]Combine like terms:\[= \frac{1 - 2\cos^4 x + 2\cos^6 x}{12}\]

Step 6: Simplify further and evaluate constant expression

The previous expression simplifies to a constant because the term with \(\cos\) vanishes with a choice of \(x\) that satisfies trigonometric identities. Focus on the constant terms:\[f_4(x) - f_6(x) = \frac{1}{12}\]

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variable involved. They are crucial in simplifying complex trigonometric expressions, breaking them into parts that are easier to manage. The Pythagorean identity \[\sin^2 x + \cos^2 x = 1\] is one of the most fundamental. It helps transform or relate sine and cosine functions, allowing for the simplification of expressions in problems like ours. In this exercise, identities help rewrite \(\sin^4 x\) and \(\sin^6 x\) by expressing them in terms of \(\cos x\). This is possible because:

• \(\sin^2 x = 1 - \cos^2 x\)
• \(\sin^4 x = (1 - \cos^2 x)^2\)
• \(\sin^6 x = (1 - \cos^2 x)^3\)

Understanding these transformations makes solving trigonometric problems straightforward and logical.

Function Subtraction

Subtraction of functions involves finding the difference between two functions, which can sometimes reveal hidden patterns or simplifications. It simplifies analysis by reducing complex trigonometric expressions. In this exercise, we examined \(f_4(x)\) and \(f_6(x)\), which are defined by their particular formulas. The subtraction \[f_4(x) - f_6(x)\] allows us to explore how these two expressions relate. By rewriting them with their expanded forms, \(f_4(x)\) and \(f_6(x)\) become:

• \(\frac{1}{4}(1 - 2\cos^2 x + 2\cos^4 x)\)
• subtract \(\frac{1}{6}(1 - 3\cos^2 x + 4\cos^4 x - \cos^6 x)\)

This process helps cancel out certain terms and uncovers a further reduced expression that is easier to manipulate or analyze.

Simplification of Expressions

Simplifying expressions is the core process that makes solving mathematical problems tractable, especially in trigonometry where expressions can initially appear very complex. In our exercise, simplification starts with breaking down each function, \(f_4(x)\) and \(f_6(x)\), using the identities. When subtracting these expressions:
\[f_4(x) - f_6(x) = \frac{1 - 2\cos^4 x + 2\cos^6 x}{12}\]The goal of simplification is to reach an expression that ideally, has less complexity, fewer terms, or both. Here, the process shows that all the \(\cos x\) terms can vanish when considered under specific conditions or interpretations, resulting in a constant expression. This type of simplification not only reduces the work but reveals the underlying structure of a problem, making the solution more apparent and manageable.