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The sine and cosine functions are fundamental in trigonometry, often appearing in a variety of mathematical contexts. One of their key relationships is expressed in the Pythagorean identity:

\[\begin{equation} \text{For any angle } A, \text{it holds that } \sin^2 A + \cos^2 A = 1. \text{(Pythagorean Identity)} \end{equation}\]
This identity implicates that the squares of these functions' values for any angle sum up to one, showing how they are intrinsically linked. Whenever you encounter a combination of \(\sin^2 A\) and \(\cos^2 A\) in an expression, consider applying this identity to simplify the expression.

Furthermore, this relationship is an indispensable tool in verifying other trigonometric identities or in solving equations involving trigonometric functions. Being familiar with this property allows for algebraic manipulation that can reveal simplified forms or alternative representations of an expression, just as seen in the exercise where expanding and combining the fractions eventually lead to applying the Pythagorean identity to simplify the numerator.

The cosecant function, denoted as \(\operatorname{cosec} A\) or \(\csc A\), is a less commonly mentioned but equally important trigonometric function. It is defined as the reciprocal of the sine function:

\[\begin{equation} \operatorname{cosec} A = \frac{1}{\sin A} \end{equation}\]
This tells us that whenever we have \(\sin A\) in the denominator of a fraction, we can express it in terms of cosecant. Learning to recognize when and how to convert between sine and cosecant can greatly assist in simplifying complex expressions, making tough-looking problems much more approachable.

In the context of the given exercise, after simplifying the combined fractions, we observed that expressing \(\frac{2}{\sin A}\) as \(2 \operatorname{cosec} A\) neatly matched the RHS of the equation, succinctly verifying the identity. This demonstrates the utility of understanding and manipulating less prominent trigonometric functions like cosecant.

Algebraic manipulation encompasses a variety of techniques used to reformulate mathematical expressions and equations into more useful or simpler forms. In the context of trigonometry, these techniques can include finding common denominators, expanding polynomials, factoring, and simplifying fractions.

In the step-by-step solution of the provided exercise, we used algebraic manipulation to transform the LHS to match the RHS. We began by finding a common denominator to combine the fractions, which is a foundational algebra skill. Next, expanding the numerator applied our ability to work with polynomial expressions, another crucial algebraic skill.

As the final steps of manipulation utilized the cancellation of common factors and knowledge of trigonometric functions, we achieved a simplified and equivalent form of the original expression. This systematic approach highlights how understanding algebra is fundamental to solving complex trigonometric equations and identities. Breaking down the process and carefully applying each stage of manipulation can support students in mastering not just individual problems but also the principles that underlie a vast array of mathematical challenges.