91Ó°ÊÓ

The square of a sum identity is a fundamental algebraic identity. It helps in simplifying expressions where two terms are added and then squared. The identity is given by:

• \[ (\text{a} + \text{b})^2 = \text{a}^2 + 2\text{ab} + \text{b}^2 \]

In this context, let \( \text{a} \) be \( \text{sin}(x) \) and \( \text{b} \) be \( \text{cos}(x) \). Applying the identity to our problem:

• \[ (\text{sin}(x) + \text{cos}(x))^2 = \text{sin}^2(x) + 2\text{sin}(x)\text{cos}(x) + \text{cos}^2(x) \]

This step breaks down the square to an expression with trigonometric terms, which helps in further simplification using other trigonometric identities. Understanding this identity and being able to apply it to trigonometric functions is a valuable skill for solving many trigonometric equations.

Another key tool in your trigonometric toolkit is the double-angle identity for sine. It helps to rewrite products of sine and cosine as a single sine of a double angle. This identity states:

• \[ \text{sin}(2x) = 2\text{sin}(x)\text{cos}(x) \]

In our problem, we see the term \( 2\text{sin}(x)\text{cos}(x) \). Using the double-angle identity, we can simplify this term to \( \text{sin}(2x) \).

• \[ \text{sin}^2(x) + 2\text{sin}(x)\text{cos}(x) + \text{cos}^2(x) = 1 + 2\text{sin}(x)\text{cos}(x) \rightarrow 1 + \text{sin}(2x) \]

After simplifying, you subtract \( \text{sin}(2x) \) from both sides, making the expression easier to simplify to \( 1 \). The double-angle identities are important as they simplify trigonometric expressions and make complex calculations manageable.

The Pythagorean identity is one of the most fundamental identities in trigonometry. It relates the square of sine and cosine to the number 1. The identity is:

• \[ \text{sin}^2(x) + \text{cos}^2(x) = 1 \]

In our exercise, this identity helps to simplify the part of the expression where \( \text{sin}^2(x) \) and \( \text{cos}^2(x) \) appear. So, when we encounter \[ (\text{sin}(x) + \text{cos}(x))^2 = \text{sin}^2(x) + 2\text{sin}(x)\text{cos}(x) + \text{cos}^2(x) \], using the Pythagorean identity, we know that \[ \text{sin}^2(x) + \text{cos}^2(x) = 1 \]. Substituting this into the expression, we get \[ 1 + 2\text{sin}(x)\text{cos}(x) \]. This simplification reduces the complexity of the expression and makes it easier to handle with additional trigonometric identities. Always remember, the Pythagorean identity is a powerful tool for simplifying trigonometric expressions!