Problem 1 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Find the values of the expression $$ \begin{aligned} &3(\sin x-\cos x)^{4}+6(\sin x+\cos x)^{2} \\ &+4\left(\sin ^{6} x+\cos ^{6} x\right) \end{aligned} $$

Step-by-Step Solution

Answer

The simplified form of the given expression is \( 7 + 12\sin^2{x}\cos^2{x} - 12\sin{x}\cos{x} \)

Step 1: Simplify the initial expression

First, rewrite the expression using the Pythagorean identity \( \sin^2{x} + \cos^2{x} = 1 \) and the application of binomial identity. We get: \( 3(\sin^2{x} + \cos^2{x} - 2\sin{x}\cos{x})^2 + 6(\sin^2{x} + \cos^2{x} + 2\sin{x}\cos{x}) + 4((\sin^2{x} + \cos^2{x})^3 - 3\sin^2{x}\cos^2{x}(\sin^2{x} + \cos^2{x})) \)

Step 2: Apply the identity

Next, apply the identity \( \sin^2{x} + \cos^2{x} = 1 \). This reduces our expression to: \( 3(1 - 2\sin{x}\cos{x})^2 + 6(1 + 2\sin{x}\cos{x}) + 4((1)^3 - 3\sin^2{x}\cos^2{x}(1)) \)

Step 3: Simplify the expression

After simplification, we reach the final expression: \( 3(1 - 2\sin{x}\cos{x})^2 + 6 + 4(1 - 3\sin^2{x}\cos^2{x}) \)

Step 4: Distribute and Combine like terms

Finally, expanding and simplifying gives: \( 3(1 - 4\sin{x}\cos{x} + 4(\sin{x}\cos{x})^2) + 6 + 4 - 12\sin^2{x}\cos^2{x} \). This simplifies further to: \( 7 + 12\sin^2{x}\cos^2{x} - 12\sin{x}\cos{x} \)

Key Concepts

Binomial Identity

The concept of the binomial identity is crucial when handling expressions involving powers of sums or differences. In simpler terms, binomial identities help extend squares, cubes, or higher powers of two terms. Among mathematicians, perhaps the most famous is the Binomial Theorem, which provides a formula to expand powers of binomials. In our expression, we see a perfect example. The term \((\sin x - \cos x)^4\) requires binomial expansion:

• The expansion uses binomial coefficients.
• It extends to expressions like \((a-b)^2 = a^2 - 2ab + b^2\) and to higher powers similarly, but with more terms.
• This increases the complexity depending on the power.

When the Pythagorean identity \( \sin^2{x} + \cos^2{x} = 1 \) is combined with binomial identities, it assists in simplifying complex trigonometric expressions like the one in the exercise, transforming convoluted terms into recognizable ones.

Pythagorean Identity

One of the fundamental identities in trigonometry is the Pythagorean identity \( \sin^2{x} + \cos^2{x} = 1 \). This identity represents the Pythagorean Theorem in a trigonometric context, relating the sine and cosine of an angle. It's an incredible tool for simplifying expressions:

• Automatically reduces any squared terms of sine and cosine to a simpler form.
• Since \( \sin^2{x} + \cos^2{x} = 1 \), substituting this in your expression helps eliminate the complexity of multiple trigonometric terms.
• For example, it turns the expression \( 3(\sin^2{x} + \cos^2{x}) \) directly into 3.

In the problem, the Pythagorean identity reduced different components, simplifying calculations by replacing combinations of sine and cosine squared terms with 1, aiding in further simplifications like factorization and expansion.

Trigonometric Simplification

Simplifying trigonometric expressions is often necessary for understanding complex equations or solving problems. This process involves reducing expressions to their simplest form using identities and algebraic manipulations. Here's how simplification works:

• Firstly, identify any applicable identities, like the Pythagorean identity.
• Apply algebraic techniques, such as expanding binomials or separating and recombining terms.
• Look for patterns that allow for replacing or reducing terms.

In this exercise, trigonometric simplification not only required applying the Pythagorean identity but also the strategic use of algebraic principles. This includes recognizing like terms and understanding the behavior of \(\sin{x}\) and \(\cos{x}\) under different operations. The result is a cleaner, more manageable expression that unveils the relationships and solutions more clearly. This expression \( 7 + 12\sin^2{x}\cos^2{x} - 12\sin{x}\cos{x} \) is one such example, where calculated simplification provides clarity.