Sure, let's solve the equation step-by-step. The equation given is:
[tex]\[ \frac{\csc x}{\sin x} - \cot^2 x = 1 \][/tex]
First, let's rewrite the trigonometric functions in terms of [tex]\(\sin(x)\)[/tex] and [tex]\(\cos(x)\)[/tex]:
1. Express [tex]\(\csc(x)\)[/tex]:
[tex]\[ \csc(x) = \frac{1}{\sin(x)} \][/tex]
2. Substitute [tex]\(\csc(x)\)[/tex] into the equation:
[tex]\[ \frac{1}{\sin(x)} \cdot \frac{1}{\sin(x)} - \cot^2(x) = 1 \][/tex]
Simplify this term:
[tex]\[ \frac{1}{\sin^2(x)} - \cot^2(x) = 1 \][/tex]
3. Express [tex]\(\cot(x)\)[/tex]:
[tex]\[ \cot(x) = \frac{\cos(x)}{\sin(x)} \][/tex]
So,
[tex]\[ \cot^2(x) = \left( \frac{\cos(x)}{\sin(x)} \right)^2 = \frac{\cos^2(x)}{\sin^2(x)} \][/tex]
4. Substitute [tex]\(\cot^2(x)\)[/tex] into the equation:
[tex]\[ \frac{1}{\sin^2(x)} - \frac{\cos^2(x)}{\sin^2(x)} = 1 \][/tex]
5. Combine the fractions:
[tex]\[ \frac{1 - \cos^2(x)}{\sin^2(x)} = 1 \][/tex]
6. Use the Pythagorean identity:
[tex]\[ \sin^2(x) + \cos^2(x) = 1 \][/tex]
So,
[tex]\[ 1 - \cos^2(x) = \sin^2(x) \][/tex]
7. Substitute this identity into the equation:
[tex]\[ \frac{\sin^2(x)}{\sin^2(x)} = 1 \][/tex]
8. Simplify the fraction:
[tex]\[ 1 = 1 \][/tex]
This simplification confirms that the equation holds true for all [tex]\( x \)[/tex] where [tex]\(\sin(x) \neq 0\)[/tex]. Therefore, the equation is an identity and valid for all values of [tex]\( x \)[/tex] for which [tex]\(\sin(x) \neq 0\)[/tex].
