Sure, let's simplify the left side of the equation [tex]\( 1 - \frac{\sin x}{\csc x} \)[/tex] and show that it is equivalent to [tex]\(\cos^2 x\)[/tex].
### Step-by-Step Solution:
1. Rewrite the left side: Start with the left side of the equation:
[tex]\[
1 - \frac{\sin x}{\csc x}
\][/tex]
2. Express [tex]\(\csc x\)[/tex]: Recall that [tex]\(\csc x = \frac{1}{\sin x}\)[/tex]. Substitute this into the equation:
[tex]\[
1 - \frac{\sin x}{\frac{1}{\sin x}}
\][/tex]
3. Simplify the fraction: Simplify the fraction inside the expression:
[tex]\[
\frac{\sin x}{\frac{1}{\sin x}} = \sin x \cdot \sin x = \sin^2 x
\][/tex]
4. Substitute back: Now the expression becomes:
[tex]\[
1 - \sin^2 x
\][/tex]
5. Use the Pythagorean identity: Recall the Pythagorean identity [tex]\(\sin^2 x + \cos^2 x = 1\)[/tex]. Solving for [tex]\(\cos^2 x\)[/tex]:
[tex]\[
\cos^2 x = 1 - \sin^2 x
\][/tex]
6. Substitute: Replace [tex]\(1 - \sin^2 x\)[/tex] with [tex]\(\cos^2 x\)[/tex]:
[tex]\[
1 - \sin^2 x = \cos^2 x
\][/tex]
The left side simplifies to [tex]\(\cos^2 x\)[/tex], which matches the right side of the equation.
### Conclusion:
Both sides of the equation [tex]\(1 - \frac{\sin x}{\csc x} = \cos^2 x\)[/tex] are indeed equal. Thus, the original equation is verified as true.
[tex]\[
1 - \frac{\sin x}{\csc x} = \cos^2 x
\][/tex]
