To simplify the given expression [tex]\(\frac{1-\sin^2(x)}{\sin(x) - \csc(x)}\)[/tex], let's follow step-by-step methods.
1. Understand the trigonometric identities:
- Recall that [tex]\(\sin(x)\)[/tex] is the sine function.
- [tex]\(\csc(x)\)[/tex] is the cosecant function, which is the reciprocal of sine. So, [tex]\(\csc(x) = \frac{1}{\sin(x)}\)[/tex].
2. Simplify the numerator:
- The expression in the numerator is [tex]\(1 - \sin^2(x)\)[/tex].
- We know the Pythagorean identity, which states: [tex]\(\sin^2(x) + \cos^2(x) = 1\)[/tex].
- Rearranging this identity gives us [tex]\(1 - \sin^2(x) = \cos^2(x)\)[/tex].
- Thus, the numerator simplifies to [tex]\(\cos^2(x)\)[/tex].
3. Simplify the denominator:
- The expression in the denominator is [tex]\(\sin(x) - \csc(x)\)[/tex].
- Substitute the definition of cosecant: [tex]\(\csc(x) = \frac{1}{\sin(x)}\)[/tex].
- Therefore, the denominator is [tex]\(\sin(x) - \frac{1}{\sin(x)}\)[/tex].
4. Combine the simplified forms:
- Now our fraction is [tex]\(\frac{\cos^2(x)}{\sin(x) - \frac{1}{\sin(x)}}\)[/tex].
5. Simplify the denominator further:
- Write [tex]\(\sin(x) - \frac{1}{\sin(x)}\)[/tex] with a common denominator: [tex]\(\frac{\sin^2(x) - 1}{\sin(x)}\)[/tex].
6. Identify and simplify the entire expression:
- Our fraction now looks like this: [tex]\(\frac{\cos^2(x)}{\frac{\sin^2(x) - 1}{\sin(x)}}\)[/tex].
- Notice that [tex]\(\sin^2(x) - 1\)[/tex] is the negative of [tex]\(1 - \sin^2(x)\)[/tex], which is [tex]\(- \cos^2(x)\)[/tex].
- Thus, we simplify the denominator: [tex]\(\frac{\sin^2(x) - 1}{\sin(x)} = \frac{-\cos^2(x)}{\sin(x)}\)[/tex].
7. Combine and simplify further:
- Our expression then becomes [tex]\(\frac{\cos^2(x)}{-\frac{\cos^2(x)}{\sin(x)}}\)[/tex], which is an equation of the form [tex]\(\frac{a}{-b/a}\)[/tex].
- Simplify by multiplying both numerator and denominator by [tex]\(\sin(x)\)[/tex]: [tex]\(\frac{\cos^2(x) \cdot \sin(x)}{-\cos^2(x)}\)[/tex].
8. Final simplification:
- Notice that [tex]\(\cos^2(x)\)[/tex] cancels out, leaving us with:
[tex]\[
\frac{\sin(x)}{-1} = -\sin(x)
\][/tex]
Therefore, the simplified form of the given expression [tex]\(\frac{1-\sin^2(x)}{\sin(x) - \csc(x)}\)[/tex] is:
[tex]\[
-\sin(x)
\][/tex]
