Problem 80 - Algebra and Trigonometry

Question

Verify the identity: $$ \sin 2 x+1=(\sin x+\cos x)^{2} $$ (Section \(6.3, \text { Examples } 3 \text { and } 6)\)

Step-by-Step Solution

Answer

After applying the double angle identity on the LHS and expanding the binomial on the RHS, the simplified LHS (\(2\sin(x)\cos(x) + 1\)) and RHS (\(1 + 2\sin(x)\cos(x)\)) are identical. Therefore, the given identity \(\sin 2 x+1=(\sin x+\cos x)^{2}\) is verified.

Step 1: Recall trigonometric identities

This problem involves dealing with trigonometric functions, therefore it is critical to recognize that \(\sin(2x) = 2\sin(x)\cos(x)\). This will be useful in the further steps.

Step 2: Simplify the LHS

Let's start by simplifying the LHS. Replace \(\sin(2x)\) with \(2\sin(x)\cos(x)\) to get \(2\sin(x)\cos(x) + 1\).

Step 3: Expand the RHS

Next, let's simplify the RHS by expanding the square of the binomial \((\sin(x) + \cos(x))^{2}\). The result is \(\sin^2(x) + 2\sin(x)\cos(x) + \cos^2(x)\). Since \(\sin^2(x) + \cos^2(x)\) is equal to 1 using the Pythagorean identity, the RHS can be simplified as \(1 + 2\sin(x)\cos(x)\).

Step 4: Compare the simplified LHS and RHS

At this point, it's evident that the simplified LHS, \(2\sin(x)\cos(x) + 1\), is equal to the simplified RHS, \(1 + 2\sin(x)\cos(x)\). Hence, the given identity is verified.

Key Concepts

Pythagorean identity

The Pythagorean identity is one of the fundamental relationships in trigonometry. It states:\[ \sin^2(x) + \cos^2(x) = 1 \]
This equation is pivotal because it holds true for any angle \(x\). It's derived from the Pythagorean Theorem and helps establish the relationship between sine and cosine for a triangle on the unit circle.

• Consider a right triangle with sides \(a\), \(b\), and hypotenuse \(c\). The Pythagorean Theorem states that \(a^2 + b^2 = c^2\).
• For the unit circle, \(c = 1\), so substituting gives \(\sin^2(x) + \cos^2(x) = 1\).
• It's useful for simplifying expressions where sine and cosine appear together. Like in the example, we can replace \(\sin^2(x) + \cos^2(x)\) with 1.

By using this identity, you can simplify complex trigonometric expressions and verify equations more efficiently.

Double angle formula

The double angle formula is a crucial tool in trigonometry used to simplify expressions involving angles twice the measure of a given angle. For sine, the formula is:\[ \sin(2x) = 2\sin(x)\cos(x) \]
This formula helps in expressing \(\sin(2x)\) using the simpler terms \(\sin(x)\) and \(\cos(x)\). It allows for easier manipulation and verification of trigonometric identities and equations.

• It's derived from the sum formula for sine, which states that \(\sin(a+b) = \sin(a)\cos(b) + \cos(a)\sin(b)\).
• By setting \(a = b = x\), you obtain \(\sin(2x) = \sin(x)\cos(x) + \cos(x)\sin(x) = 2\sin(x)\cos(x)\).
• This simplification is especially useful when solving or verifying identities, as it breaks down more complicated functions into manageable parts.

Using the double angle formula in the exercise simplifies the left-hand side, making it possible to equate it with the expanded right-hand side.

Expanding binomials

Expanding binomials is a technique in algebra that involves multiplying two binomial expressions. It's particularly useful when working with polynomial expressions and verifying identities. To expand \((a + b)^2\), you use the distributive property, leading to:\[ (a + b)^2 = a^2 + 2ab + b^2 \]
In the context of trigonometric identities, expanding allows us to express our equations in simpler terms.

• For the identity \((\sin x + \cos x)^2\), expanding results in \(\sin^2 x + 2\sin x \cos x + \cos^2 x\).
• Thanks to the Pythagorean identity, \(\sin^2 x + \cos^2 x = 1\), further simplification is possible.
• This method is essential to downloading obstacles in expressions, making problem solving more efficient and straightforward.

In the provided exercise, expanding \((\sin x + \cos x)^2\) is key to transforming the right-hand side, so it matches the left-hand side.