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Understanding trigonometric identities is crucial for solving many problems in trigonometry, including factoring trigonometric expressions. Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables involved. Examples include \(\text{Pythagorean identities}\), like \[ \sin^2(x) + \cos^2(x) = 1 \] and the \(\text{angle sum identities}\), like \[ \sin(\theta + \phi) = \sin(\theta) \cos(\phi) + \cos(\theta) \sin(\phi) \]. These identities allow you to transform and simplify trigonometric expressions, making them easier to work with and solve. Recognizing these identities during the problem-solving process can make it easier to manipulate the expression and factor it efficiently.

The grouping method is a powerful technique used in factoring polynomial and trigonometric expressions. This method involves grouping terms that share common factors, making it easier to factor the expression as a whole. In the given exercise, we first grouped the terms involving \(\text{\sin(\alpha)\cos(\alpha)}\), \(\text{\sin(\alpha)}\), \(\text{\cos(\alpha)}\), and the constant together. This resulted in the expression: \[ \sin(\alpha) \cos(\alpha) + \cos(\alpha) + \sin(\alpha) + 1 \]. After grouping, you can factor out common factors from each group. This helps break down the problem into simpler parts, paving the way for binomial factoring.

Once you have grouped the terms and factored out the common terms, the next step is to factor the common binomial. In this exercise, after using the grouping method, we arrived at: \[ \cos(\alpha)(\sin(\alpha) + 1) + 1(\sin(\alpha) + 1) \]. Notice that \(\sin(\alpha) + 1\) appears in both groups. This common binomial can now be factored out: \[ (\sin(\alpha) + 1)(\cos(\alpha) + 1) \]. Binomial factoring simplifies the expression significantly and is the final step in solving the problem. This technique can be generally applied to various mathematical problems, especially those involving expressions that can be reduced to a product of binomials.