To verify the given trigonometric identity:

$\csc^2(\theta) - \cot^2(\theta) = \cot(\theta)\tan(\theta)$

Step-by-step solution:

1. Use a fundamental trigonometric identity:

$\csc^2(\theta) - \cot^2(\theta) = 1$ This is a standard trigonometric identity.

2. Substitute the identity into the left-hand side:

The left-hand side becomes: $\csc^2(\theta) - \cot^2(\theta) = 1$

3. Analyze the right-hand side:

The right-hand side is $\cot(\theta)\tan(\theta)$. Recall the definitions of $\tan(\theta)$ and $\cot(\theta)$: $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}, \quad \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}.$ Thus: $\cot(\theta)\tan(\theta) = \left(\frac{\cos(\theta)}{\sin(\theta)}\right)\left(\frac{\sin(\theta)}{\cos(\theta)}\right).$

4. Simplify the right-hand side:

The terms $\sin(\theta)$ and $\cos(\theta)$ cancel out: $\cot(\theta)\tan(\theta) = 1.$

5. Compare both sides:

The left-hand side and the right-hand side are equal: $\csc^2(\theta) - \cot^2(\theta) = \cot(\theta)\tan(\theta).$

Conclusion:

The given equation is a valid identity.

Would you like a more detailed explanation or help with another step? Here are 5 related questions to explore further:

1. What are the fundamental trigonometric identities and how can they be derived?
2. How do we prove that $\csc^2(\theta) - \cot^2(\theta) = 1$?
3. What are the applications of verifying trigonometric identities?
4. How can the definitions of $\tan(\theta)$ and $\cot(\theta)$ be related to unit circle properties?
5. What are common mistakes made when simplifying trigonometric expressions?

Tip:

When verifying trigonometric identities, always start by recalling key fundamental identities, such as the Pythagorean identities, to simplify your work.