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The problem asks to find the indefinite integral of the following expression:

\[
\int \left( \csc(y)\cot(y) - \sec(y)\tan(y) \right) \, dy
\]

Let's break this into two separate integrals and solve each part:

1. \(\int \csc(y)\cot(y) \, dy\)
2. \(\int \sec(y)\tan(y) \, dy\)

### Step 1: Solve \(\int \csc(y)\cot(y) \, dy\)
We know that the derivative of \(-\csc(y)\) is \(\csc(y)\cot(y)\). Therefore:
\[
\int \csc(y)\cot(y) \, dy = -\csc(y)
\]

### Step 2: Solve \(\int \sec(y)\tan(y) \, dy\)
We also know that the derivative of \(\sec(y)\) is \(\sec(y)\tan(y)\). Therefore:
\[
\int \sec(y)\tan(y) \, dy = \sec(y)
\]

### Final Answer:
Combining both results, we get:
\[
\int \left( \csc(y)\cot(y) - \sec(y)\tan(y) \right) \, dy = -\csc(y) - \sec(y) + C
\]
where \(C\) is the constant of integration.

Would you like a further breakdown or more details?

Here are 5 related questions to expand understanding:

1. How can we check this result through differentiation?
2. What are some common integrals involving trigonometric functions?
3. What is the difference between indefinite and definite integrals?
4. How would the answer change if the integral were definite, say from \(a\) to \(b\)?
5. How can we use substitution to solve integrals of more complex trigonometric expressions?

**Tip**: When solving integrals, recognizing the derivative of common functions like \(\csc(x)\), \(\sec(x)\), and their combinations can save a lot of time.

Find the indefinite integral of (csc(y)cot(y) - sec(y)tan(y)) dy.

Learn how to solve the indefinite integral of (csc(y)cot(y) - sec(y)tan(y)) dy. This step-by-step solution covers key calculus concepts, including the integration of trigonometric functions. Suitable for students in Grades 11-12.

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