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The Pythagorean identity is a fundamental concept in trigonometry which states that for any angle \( x \), the square of the sine plus the square of the cosine will always equal one. Mathematically, this can be written as:
\[ \cos^2 x + \sin^2 x = 1 \]
This identity is derived from the Pythagorean theorem, which describes the relationship between the sides of a right triangle. Using HTML formatting, we can dive deeper into the components of this identity. For every right-angled triangle with hypotenuse length \( c \), and other sides of lengths \( a \) and \( b \), the Pythagorean theorem asserts that \( a^2 + b^2 = c^2 \). In trigonometry, when dealing with the unit circle (a circle with radius 1), the hypotenuse is always 1, and the lengths \( a \) and \( b \) correspond to the \( \cos \) and \( \sin \) values, respectively, for a given angle. As a result, we arrive at the Pythagorean identity.

• This identity is particularly useful when simplifying trigonometric expressions, as we can use it to convert terms involving squares of sine and cosine to simplify an equation.
• Understanding the Pythagorean identity is essential for solving a variety of trigonometric problems, including the exercise provided.

The sine double-angle identity is another key player in the realm of trigonometric identities. It expresses the sine of a doubled angle, in terms of the sine and cosine of the original angle. The identity is:
\[ \sin(2x) = 2\sin(x)\cos(x) \]
This identity is particularly useful when dealing with trigonometric problems involving expressions with \( \sin(2x) \). By understanding and utilizing this identity, you can simplify complex trigonometric expressions and resolve them into more manageable forms.This identity is derived from the sum formula for sine, \( \sin(a + b) = \sin a \cos b + \cos a \sin b \), by setting \( a = b = x \).

• Applying the sine double-angle identity allowed us to convert \( 2(\cos x \cdot \sin x) \) into \( \sin(2x) \).
• Utilizing this identity in conjunction with the Pythagorean identity was key for simplifying the expression in the exercise problem.

Algebraic identity expansion plays a fundamental role in simplifying and solving mathematical equations. One of the most frequently used algebraic identities is the square of a binomial:
\[ (a + b)^2 = a^2 + 2ab + b^2 \]
This formula is key when expanding expressions such as \( (\cos x + \sin x)^2 \) as it directly leads us to the first step in solving many trigonometric problems. The expansion lays out the foundation for further simplification using trigonometric identities like the Pythagorean and double-angle sine identities.

• The first step we took in solving the provided exercise was expanding the left side using this identity.
• We recognized that \( a \) and \( b \) from the algebraic identity can represent any trigonometric function, including \( \cos x \) and \( \sin x \) from our exercise.

In this context, mastering the use of algebraic identity expansion is crucial as it allows us to rewrite and simplify complex expressions into something that is easier to work with using other identities, ultimately leading to a solution.