Problem 40 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The identity is verified as both sides simplify to the same expression.

Step 1: Expand Left Side of the Equation

Begin by expanding the left side of the identity: \[(\sin x + \cos x)^4.\]This can be expanded by first squaring the binomial twice. First, expand \[(\sin x + \cos x)^2 = \sin^2 x + 2\sin x\cos x + \cos^2 x.\]Recognize that \(\sin^2 x + \cos^2 x = 1\), thus the equation becomes:\[1 + 2\sin x\cos x.\]Now, square this result:\[(1 + 2\sin x\cos x)^2.\]

Step 2: Expand Right Side of the Equation

The right side of the identity is \[(1 + 2\sin x\cos x)^2.\]Expanding this gives:\[1^2 + 2\cdot1\cdot2\sin x\cos x + (2\sin x\cos x)^2.\]So this simplifies to:\[1 + 4\sin x\cos x + 4\sin^2 x\cos^2 x.\]

Step 3: Simplify Both Sides Further

Recall from the Pythagorean identity that \(\sin^2 x + \cos^2 x = 1\). Find an equivalent expression for \(\sin^2 x\cos^2 x\):\[\sin^2 x\cos^2 x = \left(\frac{\sin(2x)}{2}\right)^2 = \frac{1}{4} \sin^2(2x).\]Thus \(\4\sin^2 x\cos^2 x = \sin^2(2x),\)leading to:\[1 + 4\sin x\cos x + \sin^2(2x).\]

Step 4: Compare and Verify Equality

Both the expanded forms of \((\sin x + \cos x)^4\) and \((1 + 2\sin x\cos x)^2\) simplify to the same expression:\[1 + 4\sin x\cos x + 4\sin^2 x\cos^2 x.\]Hence, the identity is verified: \[(\sin x + \cos x)^4 = (1 + 2 \sin x \cos x)^2.\]

Key Concepts

Pythagorean Identity

The Pythagorean identity is a fundamental relation in trigonometry, linking the sine and cosine of an angle. The identity states that for any angle \(x\), \(\sin^2 x + \cos^2 x = 1\). This relationship is derived from the Pythagorean theorem applied to the unit circle, where the radius is 1. In the context of the problem, this identity helps simplify trigonometric expressions.

• When you have \(\sin x\) and \(\cos x\) terms, noticing the Pythagorean identity can transform these into simpler or more familiar forms, as
  seen in Step 1 when \((\sin^2 x + \cos^2 x)\) is replaced by 1.
• This property not only simplifies expressions but also verifies identities
  by comparing both sides.

Understanding and spotting \(\sin^2 x + \cos^2 x\) together can make handling trigonometric problems more efficient.

Trigonometric Expansion

Trigonometric expansion involves breaking down expressions involving sums or products of trigonometric functions. It is a useful tool in forming or validating identities. In our exercise, we expanded expressions like \((\sin x + \cos x)^2\) and \((1 + 2 \sin x \cos x)^2\).

• For \((\sin x + \cos x)^2\), expanding yields \(\sin^2 x + 2\sin x\cos x + \cos^2 x\).
• For the right side, \((1 + 2 \sin x \cos x)^2\) expands to \(1 + 4\sin x\cos x + 4\sin^2 x\cos^2 x\).

This technique aids in verifying trigonometric identities by transforming complex equations into simpler or more recognizable forms. Understanding the methodical expansion is vital to grasp more extensive formulas or identities in trigonometry.

Binomial Theorem

The binomial theorem provides a powerful way to expand expressions that are raised to any power, appearing frequently in algebra and calculus. It applies neatly to binomials, which are expressions with two terms, like \((a + b)^n\).

• The formula is \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\), where \(\binom{n}{k}\) are the binomial coefficients.
• In the exercise, it simplifies \(\sin x + \cos x\) and \(1 + 2\sin x \cos x\), showing how expressions can be decomposed systematically.

The theorem guarantees a methodical approach to expanding and simplifying powers of binomials, essential in deriving or proving identities. Having a strong grip on the binomial theorem can immensely simplify handling rising powers in trigonometric problems.