Question - Simplifying Trigonometric Expression
Solution:
To solve the expression, we need to simplify it by using trigonometric identities where possible. The expression is:\[ \frac{\csc(x)}{1 - \cos^2(x)} - \csc^3(x) \]We can recognize that $$\csc(x) = \frac{1}{\sin(x)}$$, and using the Pythagorean identity $$1 - \cos^2(x) = \sin^2(x)$$, we can rewrite the first term:\[ \frac{\csc(x)}{1 - \cos^2(x)} = \frac{\frac{1}{\sin(x)}}{\sin^2(x)} = \frac{1}{\sin^3(x)} \]Now we rewrite $$\csc^3(x)$$ as:\[ \csc^3(x) = \left(\frac{1}{\sin(x)}\right)^3 = \frac{1}{\sin^3(x)} \]With these substitutions, the expression simplifies to:\[ \frac{1}{\sin^3(x)} - \frac{1}{\sin^3(x)} \]Now it is clear that both terms are the same and they cancel each other out. Thus, the simplified expression equals zero.\[ \frac{\csc(x)}{1 - \cos^2(x)} - \csc^3(x) = 0 \]
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