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In trigonometry, simplifying expressions can help us understand the problem better and find solutions more easily. Simplifying involves using known identities and algebraic techniques to make an expression simpler.

There are many trig identities we should know, such as Pythagorean identities, reciprocal identities, and angle sum and difference identities. In the given exercise, we use the identity for the sine function, which is \( \sin(-x) = -\sin(x) \).

Applying this to the expression \( \frac{\sin^2(-x) - \sin(-x)}{1 - \sin(-x)} \), we first substitute the identity into the expression. This gives a more simplified form that is easier to work with.

Keep practicing such substitutions to get better at simplifying complex trigonometric expressions.

Understanding the properties of the sine function is crucial in trigonometry. One key property is that the sine function is odd. This means that \( \sin(-x) = -\sin(x) \).

This property helps us simplify expressions involving negative angles. It also makes solving trigonometric equations more manageable.

Another important property of the sine function is its range. The values of \( \sin(x) \) always lie between -1 and 1 for all real numbers x.

When dealing with trigonometric expressions, these properties allow us to transform and simplify the expressions more effectively.

Factoring is finding terms that multiply to make a given expression. In trigonometry, factoring can help simplify and solve equations.

In our example, we had to factor the numerator \( \sin^2(x) + \sin(x) \). This can be factored into \( \sin(x)(\sin(x) + 1) \).

This factoring step is critical as it allows cancellation with terms in the denominator. After factoring, we can cancel common factors in the numerator and denominator to further simplify the expression to \( \sin(x) \).

Practicing factoring will improve problem-solving skills in algebra and trigonometry.