Answer :

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]