To verify or simplify the equation $\csc \theta - \sin \theta = \cot \theta \cos \theta$, let's start by rewriting each trigonometric term in terms of sine and cosine.

Step 1: Rewrite in terms of sine and cosine

1. Rewrite $\csc \theta$ as $\frac{1}{\sin \theta}$.
2. Rewrite $\cot \theta$ as $\frac{\cos \theta}{\sin \theta}$.

This gives us: $\frac{1}{\sin \theta} - \sin \theta = \frac{\cos \theta}{\sin \theta} \cdot \cos \theta$

Step 2: Simplify the Left Side

The left side becomes: $\frac{1}{\sin \theta} - \sin \theta = \frac{1 - \sin^2 \theta}{\sin \theta}$

Using the Pythagorean identity, $1 - \sin^2 \theta = \cos^2 \theta$, this simplifies to: $\frac{\cos^2 \theta}{\sin \theta}$

Step 3: Simplify the Right Side

The right side is: $\frac{\cos \theta}{\sin \theta} \cdot \cos \theta = \frac{\cos^2 \theta}{\sin \theta}$

Step 4: Conclusion

Both sides are equal: $\frac{\cos^2 \theta}{\sin \theta} = \frac{\cos^2 \theta}{\sin \theta}$

Thus, the original equation holds: $\csc \theta - \sin \theta = \cot \theta \cos \theta$

Would you like further clarification on any steps?

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Here are five related questions to explore:

1. How can we derive other trigonometric identities from basic identities like $\sin^2 \theta + \cos^2 \theta = 1$?
2. What are the proofs for other fundamental trigonometric identities?
3. How do reciprocal identities relate to trigonometric simplifications?
4. Can you solve equations involving mixed trigonometric terms (like $\sin \theta + \cos \theta$)?
5. How do transformations of trigonometric functions affect their graphs?

Tip: When working with trigonometric identities, converting terms to sine and cosine is often helpful for simplification.