We aim to verify the identity: $\frac{\cot^2 t}{\csc t} = \csc t - \sin t$

Part 1: Start with the Left-Hand Side (LHS)

The LHS is: $\frac{\cot^2 t}{\csc t}.$

Step 1: Express $\cot^2 t$ using a Pythagorean identity.

Using $\cot^2 t = \csc^2 t - 1$: $\frac{\cot^2 t}{\csc t} = \frac{\csc^2 t - 1}{\csc t}.$

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Part 2: Simplify the fraction

Step 2: Separate the fraction into two terms.

$\frac{\csc^2 t - 1}{\csc t} = \frac{\csc^2 t}{\csc t} - \frac{1}{\csc t}.$

Simplify each term: $\frac{\csc^2 t}{\csc t} = \csc t, \quad \frac{1}{\csc t} = \sin t.$

Thus: $\frac{\cot^2 t}{\csc t} = \csc t - \sin t.$

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Verification Complete

Since the transformed LHS equals the RHS: $\csc t - \sin t = \csc t - \sin t,$ the identity is verified.

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Summary of Steps

1. Apply the Pythagorean identity: $\cot^2 t = \csc^2 t - 1$.
2. Rewrite the fraction: $\frac{\csc^2 t - 1}{\csc t}$.
3. Separate into two terms: $\frac{\csc^2 t}{\csc t} - \frac{1}{\csc t}$.
4. Simplify to $\csc t - \sin t$.

Would you like further clarifications or additional examples? Here are some related questions to consider:

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Related Questions:

1. How do Pythagorean identities relate to trigonometric transformations?
2. Can you verify similar identities, such as $\cot^2 t + 1 = \csc^2 t$?
3. What are the reciprocal identities for $\sin t$, $\cos t$, and $\tan t$?
4. How do you approach verifying identities systematically?
5. Can you provide additional practice examples for verifying trigonometric identities?

Tip:

Always start with the more complex side when verifying identities to simplify toward the simpler side.