Problem 57 - Contemporary Precalculus

Question

Simplify the given expression. $$(\sin x+\cos x)^{2}-\sin 2 x$$

Step-by-Step Solution

Answer

Answer: The simplified form of the given expression is $$1$$.

Step 1: Expand the square term

To simplify the expression, first expand the square of the sum: $$(\sin x + \cos x)^2 = (\sin x)^2 + 2\sin x\cos x + (\cos x)^2$$

Step 2: Apply the Pythagorean identity

Remember that one of the Pythagorean identities is: $$\sin^2 x + \cos^2 x = 1$$ In the expanded expression, the terms \((\sin x)^2 + (\cos x)^2\) can be substituted with \(1\): $$1 + 2\sin x\cos x$$

Step 3: Use the double angle identity for sine

The double angle identity for sine is: $$\sin 2x = 2\sin x\cos x$$ We see that the term \(2\sin x\cos x\) in our expression is equal to \(\sin 2x\). Therefore, the expression becomes: $$1 + \sin 2x$$

Step 4: Combine terms with the original expression

Now, combine our simplified expression with the original expression by subtracting \(\sin 2x\): $$(\sin x + \cos x)^2 - \sin 2x = (1 + \sin 2x) - \sin 2x = 1$$ The simplified expression is a constant: $$1$$.

Key Concepts

Pythagorean Identity

The Pythagorean Identity is one of the most fundamental trigonometric identities. It states that for any angle \( x \), the square of the sine of \( x \) plus the square of the cosine of \( x \) is always equal to 1. Recall:\[sin^2 x + cos^2 x = 1\]This identity is derived from the Pythagorean Theorem applied to a unit circle. In this context, the radius of the circle is 1, hence any point \((\cos x, \sin x)\) on the circle satisfies the equation.

• When using this identity, you're essentially substituting \((\sin x)^2 + (\cos x)^2\) with 1.
• This simplification is helpful in many trigonometric problems, including simplifying expressions with squares of sine and cosine terms.

Double Angle Identity

The Double Angle Identity for sine expresses \( \sin 2x \) in terms of \( \sin x \) and \( \cos x \). The formula is: \[ sin 2x = 2\sin x\cos x\]This identity is particularly useful when dealing with expressions that have products of sine and cosine terms.

• It helps us recognize and replace expressions like \( 2\sin x\cos x \) with a simpler \( \sin 2x \).
• Using this identity, we can simplify complex expressions and solve trigonometric equations efficiently.

In the given problem, noticing \( 2\sin x\cos x \) allows direct use of this identity to simplify the expression.

Expression Simplification

Expression simplification involves reducing a trigonometric expression to its simplest form. This process often utilizes trigonometric identities, like the ones we've discussed, to eliminate complicated terms and replace them with simpler ones.

• The goal is to find an equivalent expression that's easier to interpret or evaluate.
• In our exercise, we used both the Pythagorean Identity and the Double Angle Identity to transform the given expression step-by-step.

The steps are typically:1. **Expand** any squared sums or products.2. **Substitute** known identities to reduce the expression.3. **Combine** like terms or simplify further by canceling out terms wherever possible.For example, starting with \((\sin x + \cos x)^2 - \sin 2x\), you first expand the squared term. Then use identities to replace parts of the expression, and finally simplify by combining terms to get a concise result, like the final answer \(1\) in our case.