91Ó°ÊÓ

The Pythagorean Identity is a fundamental concept in trigonometry. It tells us that for any angle \(x\), the sum of the square of sine and the square of cosine equals one. Mathematically, it is expressed as:

• \( \sin^2 x + \cos^2 x = 1 \)

This identity is derived from the Pythagorean Theorem applied in the unit circle, where the radius is one, and the coordinates of any point can be considered as \((\cos x, \sin x)\).
These points form a right triangle with the circle's center and the circle's radius. Since the radius is the hypotenuse and equals one, the other two sides, \( \cos x \) and \( \sin x \), satisfy the identity.
In the problem we solved, we utilized the Pythagorean Identity to replace \( \sin^2 x + \cos^2 x \) with 1, simplifying our expression significantly.

The Binomial Expansion is a key algebraic tool when dealing with expressions raised to a power. It helps us expand and simplify expressions like \((a + b)^2\). According to the Binomial Theorem, the formula for squaring a binomial is:

• \( (a + b)^2 = a^2 + 2ab + b^2 \)

We apply this concept to expand trigonometric expressions. In the original exercise, our terms were \(a = \sin x\) and \(b = \cos x\). Thus:

• \( (\sin x + \cos x)^2 = \sin^2 x + 2\sin x \cos x + \cos^2 x \)

Expanding the binomial lays the groundwork for further simplification using identities like the Pythagorean Identity.
This step is crucial for reducing complex expressions to a more manageable form.

Simplifying trigonometric expressions is essential for making calculations more manageable and for solving trigonometric equations. This involves using various identities, such as the Pythagorean Identity, to combine and reduce terms.
In the problem, we expanded \((\sin x + \cos x)^2\) using the Binomial Expansion. After expansion, we had:

• \( \sin^2 x + 2\sin x \cos x + \cos^2 x \)

By applying the Pythagorean Identity, \( \sin^2 x + \cos^2 x = 1 \), this expression was simplified to:

• \( 1 + 2\sin x \cos x \)

The result is a much simpler expression that is easier to work with. This simplification helps in solving equations and computing integrals or derivatives in calculus.