Prove the Identity $\frac{\sin ^{3} A-\cos ^{3} A}{\sin ^{2} A-\cos ^{2} A}=\frac{2+\sin 2 A}{2(\sin A+\cos A)}$

question 21

Essay

Prove the identity:
$\frac{\sin ^{3} A-\cos ^{3} A}{\sin ^{2} A-\cos ^{2} A}=\frac{2+\sin 2 A}{2(\sin A+\cos A)}$

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Definitions:

Borderline Personality

A mental health disorder characterized by unstable moods, behaviors, and relationships, with a marked difficulty in managing emotions and impulses.

Narcissistic Personality Disorder

A mental condition characterized by a long-term pattern of exaggerated self-importance, a need for excessive attention and admiration, and a lack of empathy for others.

Ego-dystonic

Refers to troubling thoughts, feelings, beliefs, or behaviors that one experiences as alien or foreign and would like to be rid of.

Prevalence

The proportion of a population found to have a particular condition or characteristic at a specific time.