Perform each indicated operation and simplify the result so that there are no quotients.
1/( sin α - 1) - 1/(sin α + 1)

Verified step by step guidance

1. Identify a common denominator for the two fractions. The common denominator is \((\sin \alpha - 1)(\sin \alpha + 1)\).
2. Rewrite each fraction with the common denominator: \(\frac{1}{\sin \alpha - 1} = \frac{\sin \alpha + 1}{(\sin \alpha - 1)(\sin \alpha + 1)}\) and \(\frac{1}{\sin \alpha + 1} = \frac{\sin \alpha - 1}{(\sin \alpha - 1)(\sin \alpha + 1)}\).
3. Subtract the two fractions: \(\frac{\sin \alpha + 1}{(\sin \alpha - 1)(\sin \alpha + 1)} - \frac{\sin \alpha - 1}{(\sin \alpha - 1)(\sin \alpha + 1)}\).
4. Combine the numerators over the common denominator: \(\frac{(\sin \alpha + 1) - (\sin \alpha - 1)}{(\sin \alpha - 1)(\sin \alpha + 1)}\).
5. Simplify the numerator: \((\sin \alpha + 1) - (\sin \alpha - 1) = 2\), resulting in \(\frac{2}{(\sin \alpha - 1)(\sin \alpha + 1)}\).

Key Concepts

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, relate angles to the ratios of sides in right triangles. In this question, the sine function is used, which is defined as the ratio of the length of the opposite side to the hypotenuse in a right triangle. Understanding how these functions behave and their properties is essential for manipulating expressions involving them.

Common Denominator

When performing operations with fractions, finding a common denominator is crucial for combining them. In this case, the two fractions have different denominators: (sin α - 1) and (sin α + 1). To simplify the expression, one must find a common denominator, which is the product of the two denominators, allowing for the combination of the fractions into a single expression.

Simplification of Expressions

Simplification involves reducing an expression to its simplest form, often by eliminating common factors or combining like terms. In this problem, after finding a common denominator and combining the fractions, further simplification may involve factoring or canceling terms. This process is essential to ensure the final result is presented without quotients, as specified in the question.