Problem 42 - Algebra and Trigonometry

Question

Verify the identity. $$ (\sin x+\cos x)^{4}=(1+2 \sin x \cos x)^{2} $$

Step-by-Step Solution

Answer

Both sides are equivalent; the identity is verified.

Step 1: Expand the Left Side

First, expand the expression \((\sin x + \cos x)^4\). Start by recognizing it as \((a+b)^4\), where \(a = \sin x\) and \(b = \cos x\). The expansion of \((a+b)^4\) using the binomial theorem is \(a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4\). Substitute back \(a\) and \(b\):\[(\sin x)^4 + 4(\sin x)^3(\cos x) + 6(\sin x)^2(\cos x)^2 + 4(\sin x)(\cos x)^3 + (\cos x)^4.\]

Step 2: Simplify Each Term

Now, simplify each term in the expanded expression:- \((\sin x)^4 + (\cos x)^4\) remains unchanged.- The term \(6(\sin x)^2(\cos x)^2\) can be rewritten as \(6 \sin^2 x \cos^2 x\).- The terms \(4(\sin x)^3(\cos x)\) and \(4(\sin x)(\cos x)^3\) combined give \(4 \sin x \cos x (\sin^2 x + \cos^2 x)\). Since \(\sin^2 x + \cos^2 x = 1\), this simplifies to \(4 \sin x \cos x\).

Step 3: Expand the Right Side

Now, expand the expression \((1 + 2 \sin x \cos x)^2\) using the formula for squaring a binomial:\(a^2 + 2ab + b^2\), where \(a=1\) and \(b=2\sin x\cos x\). This yields: \[1 + 4 \sin x \cos x + 4 (\sin x \cos x)^2.\] Simplifying, the last term becomes \(4 \sin^2 x \cos^2 x\).

Step 4: Compare Both Sides

Compare the expressions obtained in Step 2 and Step 3:- From Step 2, we have \((\sin x)^4 + 4 \sin x \cos x + 6 \sin^2 x \cos^2 x + (\cos x)^4\).- From Step 3, the simplified form is \(1 + 4 \sin x \cos x + 4 \sin^2 x \cos^2 x\).Recognize that \((\sin x)^4 + (\cos x)^4 + 2(\sin^2 x \cos^2 x) = 1\), which is a trigonometric identity. Thus, both sides are equivalent.

Key Concepts

Binomial Theorem

The Binomial Theorem is a powerful tool in algebra used to expand expressions raised to a power, such as \((a + b)^n\). This theorem provides a way to write these expressions as a sum of individual terms, each consisting of a specific combination of the elements involved. In this exercise, the left side of the equation \((\sin x + \cos x)^4\) is the focus.We identify it as \((a + b)^4\) with the components \(a = \sin x\) and \(b = \cos x\). According to the Binomial Theorem, \((a+b)^4\) can be expanded as:

• \(a^4\)
• \(4a^3b\)
• \(6a^2b^2\)
• \(4ab^3\)
• \(b^4\)

This expansion process involves applying coefficients from Pascal's Triangle or using the Binomial Coefficient, \( \binom{n}{k} \), which calculates how many ways you can pick \(k\) unordered outcomes from \(n\) possibilities.After applying these coefficients and substituting back \(\sin x\) and \(\cos x\) into the formula, we get a polynomial expression that's more manageable. The Binomial Theorem simplifies complex expressions, making them easier to work with.

Trigonometric Functions

Trigonometric functions like sine and cosine are fundamental in math and describe relationships in angles and circles. In this identity verification, we encounter functions \[(\sin x)^4\] and \[(\cos x)^4\].Trigonometric identities, such as \(\sin^2 x + \cos^2 x = 1\), help us simplify expressions and verify identities. This specific identity underpins each simplification step and enables us to express certain terms more simply. For example, in our exercise, it allows the reduction of \[4\sin x \cos x(\sin^2 x + \cos^2 x)\] to just \[4\sin x \cos x\].These functions are not only crucial for calculating angles and side lengths in geometry but also play a vital role in transforming and solving algebraic equations, as illustrated here. Understanding how to manipulate these functions and their identities is essential for problem-solving.

Expansion and Simplification

The process of expansion and simplification bridges complex equations to more understandable forms. Our exercise demands this approach for both sides of the equation to verify equivalence.Starting with the expansion, we systematically break down complex expressions:

• For \((\sin x + \cos x)^4\), apply the Binomial Theorem to achieve a polynomial involving different powers of \(\sin x\) and \(\cos x\).
• On the other side, \((1 + 2\sin x \cos x)^2\) uses the basic formula for squaring a binomial: \(a^2 + 2ab + b^2\).

The next phase is simplification, where each term in both expansions is reduced to its lowest form, primarily through applying trigonometric identities.Such as:

• \((\sin x)^4 + (\cos x)^4\) becomes part of a known identity involving \(\sin x\) and \(\cos x\).
• Terms like \(6\sin^2 x \cos^2 x\) align with expanded and simplified forms on both sides.

By comparing each component of the expanded forms, subtle equivalences affirm that both sides of the equation are indeed equal. This approach highlights the importance of mastering such expansions and reductions to solve and verify mathematical expressions effectively.