https://file.aiquickdraw.com/mathbot/file/1725516625721z3qcwuc9.jpg

The equation provided in the image is:

\[
\frac{\cos \theta + \sin \theta - \sin^3 \theta}{\sin \theta} = \cot \theta + \cos^2 \theta
\]

Let’s solve this step by step.

### Left-hand side (LHS):
\[
\frac{\cos \theta + \sin \theta - \sin^3 \theta}{\sin \theta}
\]
We can split this into separate terms:
\[
\frac{\cos \theta}{\sin \theta} + \frac{\sin \theta}{\sin \theta} - \frac{\sin^3 \theta}{\sin \theta}
\]

Simplify each term:
\[
\frac{\cos \theta}{\sin \theta} = \cot \theta, \quad \frac{\sin \theta}{\sin \theta} = 1, \quad \frac{\sin^3 \theta}{\sin \theta} = \sin^2 \theta
\]

So, the LHS becomes:
\[
\cot \theta + 1 - \sin^2 \theta
\]

### Right-hand side (RHS):
\[
\cot \theta + \cos^2 \theta
\]

### Now compare both sides:
We already have \(\cot \theta\) on both sides. We need to show:
\[
1 - \sin^2 \theta = \cos^2 \theta
\]
This is true by the Pythagorean identity:
\[
\sin^2 \theta + \cos^2 \theta = 1
\]

Thus, the equation is correct and both sides are equal.

---

Would you like more details or have any questions about the steps?

Here are some related questions for practice:

1. Can you derive the identity \(\sin^2 \theta + \cos^2 \theta = 1\)?
2. What is the relationship between \(\cot \theta\) and \(\tan \theta\)?
3. How do you convert trigonometric expressions into terms of sine and cosine only?
4. What are some other common Pythagorean identities in trigonometry?
5. How can you simplify trigonometric fractions using basic identities?

**Tip:** When verifying identities, break down complex terms into simpler expressions involving basic trigonometric functions like \(\sin \theta\) and \(\cos \theta\).

Simplify and verify the equation: (cos θ + sin θ - sin³ θ) / sin θ = cot θ + cos² θ

Learn how to simplify and verify the trigonometric identity (cos θ + sin θ - sin³ θ) / sin θ = cot θ + cos² θ. This problem involves understanding basic trigonometric concepts, identities, and their properties, suitable for students in Grades 10-12.

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