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In trigonometry and algebra, the sum of cubes is a formula used to factor expressions where two terms, each raised to the power of three, are added together. This formula is expressed as:\[ a^3 + b^3 = (a+b)(a^2 - ab + b^2) \]In this exercise, the formula is applied to factor trigonometric expressions like \( \cos^3 \theta + \sin^3 \theta \). This formula helps simplify the original expression by recognizing that the numerator can be rewritten in a product form.To effectively use the sum of cubes formula:

• Identify terms that match the form \( a^3 + b^3 \).
• Assign \( a = \cos \theta \) and \( b = \sin \theta \).
• Apply the sum of cubes formula to rewrite the expression.

By doing so, we transformed a complex fraction into a simpler form. This is a great example of using algebraic identities to simplify trigonometric problems.

The Pythagorean Identity is a crucial element in trigonometry, linking the square of sine and cosine functions to the number 1:\[ \cos^2 \theta + \sin^2 \theta = 1 \]This identity is derived from the Pythagorean theorem and underlies many simplifications in trigonometry.In our problem, after factoring with the sum of cubes formula, we end up with a term \( \cos^2 \theta + \sin^2 \theta \), which equals 1 according to the Pythagorean Identity. This simple substitution significantly simplifies the expression.Steps to apply the Pythagorean Identity:

• Identify the parts of the expression that can be rewritten using the identity, \( \cos^2 \theta + \sin^2 \theta \).
• Replace \( \cos^2 \theta + \sin^2 \theta \) with 1.

Using this identity is often a straightforward solution to making trigonometric expressions more manageable, especially in proving identities or in calculations involving integration and differentiation.

The double angle identity for sine provides a useful relationship for manipulating expressions involving products of sine and cosine. Specifically, it states:\[ \sin 2\theta = 2 \sin \theta \cos \theta \]This identity expresses the sine of double the angle in terms of functions of \( \theta \).Within our exercise, after applying the Pythagorean Identity, we are left with \( 1 - \cos \theta \sin \theta \). By recognizing that \( \cos \theta \sin \theta \) can be written as \( \frac{1}{2} \sin 2\theta \) using the double angle identity for sine, the problem simplifies even further.When using the double angle identity:

• Note when a trigonometric expression involves \( \sin \theta \cos \theta \).
• Use \( \sin 2\theta = 2 \sin \theta \cos \theta \) to express products like \( \cos \theta \sin \theta \) as \( \frac{1}{2} \sin 2\theta \).
• Substitute and simplify accordingly.

The double angle identity is especially helpful for reducing longer expressions into simpler, equivalent forms, proving identities, and solving equations.