91Ó°ÊÓ

To understand how to rewrite trigonometric expressions, it's crucial to grasp the concept of the sum-to-product identity.
This identity transforms the sum or difference of two sines or cosines into a product of sines and cosines.
For example, the sum-to-product identities are defined as:
\[ \begin{aligned} \text{1. } \sin A + \sin B &= 2 \sin \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right) \end{aligned} \]
So, if we have \sin(2(\alpha + \beta)) \, we apply the identity:
\[ \sin(2 \alpha) + \sin(2 \beta) = 2 \sin(\alpha + \beta) \cos(\alpha - \beta) \. \]
This explains how rewriting sums of sine functions can often simplify complex trigonometric equations.

The angle sum identity is a fundamental tool in trigonometry for dealing with angles that sum to another significant value.
Specifically, the angle sum identity for sine is given by:
\[ \sin(2(\pi - x)) = \sin(2x) \. \]
In our problem, knowing that \alpha + \beta + \gamma = \pi \ implies:
\[ \gamma = \pi - (\alpha + \beta) \].
Applying the identity to \sin(2(\pi - (\alpha + \beta))) = \sin(2(\alpha + \beta)) \,
helps in rewriting \sin(2 \gamma) \ applying our fundamental trigonometric identity principles.

The product-to-sum identity converts products of trigonometric functions into sums or differences.
This is particularly useful when handling terms like \[ \sin A \sin B, \] \cos A \cos B \, and \sin A \cos B \. The key product-to-sum identities can be expressed as:
\[ \begin{aligned} \sin A \sin B &= \frac{1}{2} [\cos(A-B) - \cos(A+B)] \end{aligned} \]
For instance, in the problem, after substituting using the sum-to-product identity, we get \[ \sin(2 \alpha) + \sin(2 \beta) + 2 \sin(\alpha + \beta) \cos(\alpha + \beta) \].
Now apply the product-to-sum identity as:
\[ 2 \sin(x + y) \cos(x - y) = \sin(2x) + \sin(2y). \]
This helps simplify and combine all the terms, guiding us towards the goal.
Understanding how to apply the product-to-sum identities is crucial in achieving this final result.