91Ó°ÊÓ

Trigonometric identities are equations that involve trigonometric functions and are true for every value of the variable within a certain range. These identities are crucial for simplifying and solving trigonometric expressions.

Some common trigonometric identities include the Pythagorean identity, the sum and difference identities, and the double angle identities.

In this exercise, we use the basic trigonometric function \( \sin \alpha \), which represents the sine of angle alpha. It's essential to recognize these trigonometric functions and identities to manipulate and simplify expressions involving angles.

The difference of squares is a specific algebraic identity used frequently in mathematics. The formula for the difference of squares is: \[ (a + b)(a - b) = a^2 - b^2. \]

In the given problem, we see: \[ (\sin \alpha + 2)(\sin \alpha - 2). \]

We can identify this as a difference of squares where \( a = \sin \alpha \) and \( b = 2 \).

Applying the difference of squares formula, we substitute \( \sin \alpha \) for \( a \) and \( 2 \) for \( b \), resulting in:

\( (\sin \alpha)^2 - 2^2. \)

Simplifying expressions involves reducing them to their simplest form. This is achieved by performing operations and applying identities accurately.

In the problem, after applying the difference of squares identity, we are left with:
\[ (\sin \alpha)^2 - 4. \]
Here’s how we simplify it:
\( (\sin \alpha)^2 \) remains as is since it's already in the simplest form.
However, \( 2^2 \) simplifies to \ 4 \.

Thus, the final simplified expression is:
\[ (\sin \alpha)^2 - 4. \]

Understanding these steps helps to see how applying trigonometric identities and other algebraic techniques can simplify otherwise complicated expressions. Breaking down problems into smaller parts can make them more manageable.