Prove that sin²AtanA+cos²AcotA+2sinAcosA= tanA+cotA

We are given the expression to prove:

\[
\sin^2(A) \tan(A) + \cos^2(A) \cot(A) + 2\sin(A)\cos(A) = \tan(A) + \cot(A)
\]

### Step-by-Step Proof

1. **Rewrite the left-hand side (LHS) of the equation** using the definitions of \(\tan(A)\) and \(\cot(A)\):

\[
\tan(A) = \frac{\sin(A)}{\cos(A)} \quad \text{and} \quad \cot(A) = \frac{\cos(A)}{\sin(A)}
\]

Thus, the left-hand side becomes:

\[
\sin^2(A) \cdot \frac{\sin(A)}{\cos(A)} + \cos^2(A) \cdot \frac{\cos(A)}{\sin(A)} + 2\sin(A)\cos(A)
\]

Simplify each term individually:

\[
\sin^2(A) \cdot \frac{\sin(A)}{\cos(A)} = \frac{\sin^3(A)}{\cos(A)}
\]

\[
\cos^2(A) \cdot \frac{\cos(A)}{\sin(A)} = \frac{\cos^3(A)}{\sin(A)}
\]

\[
2\sin(A)\cos(A) = 2\sin(A)\cos(A)
\]

So the LHS becomes:

\[
\frac{\sin^3(A)}{\cos(A)} + \frac{\cos^3(A)}{\sin(A)} + 2\sin(A)\cos(A)
\]

2. **Combine the fractions** for the first two terms:

To combine \(\frac{\sin^3(A)}{\cos(A)}\) and \(\frac{\cos^3(A)}{\sin(A)}\), we take the common denominator:

\[
\frac{\sin^3(A)}{\cos(A)} + \frac{\cos^3(A)}{\sin(A)} = \frac{\sin^4(A) + \cos^4(A)}{\sin(A)\cos(A)}
\]

Thus, the LHS now becomes:

\[
\frac{\sin^4(A) + \cos^4(A)}{\sin(A)\cos(A)} + 2\sin(A)\cos(A)
\]

3. **Simplify the expression** by adding the terms:

We can express the sum of terms as:

\[
\frac{\sin^4(A) + \cos^4(A) + 2\sin^2(A)\cos^2(A)}{\sin(A)\cos(A)}
\]

Notice that:

\[
\sin^4(A) + \cos^4(A) + 2\sin^2(A)\cos^2(A) = (\sin^2(A) + \cos^2(A))^2
\]

Since \(\sin^2(A) + \cos^2(A) = 1\), we have:

\[
(\sin^2(A) + \cos^2(A))^2 = 1^2 = 1
\]

So the LHS simplifies to:

\[
\frac{1}{\sin(A)\cos(A)}
\]

4. **Simplify the final expression**:

\[
\frac{1}{\sin(A)\cos(A)} = \frac{1}{\sin(A)} + \frac{1}{\cos(A)} = \cot(A) + \tan(A)
\]

Thus, the left-hand side (LHS) is equal to the right-hand side (RHS):

\[
\boxed{\tan(A) + \cot(A)}
\]

### Conclusion:

We have successfully shown that:

\[
\sin^2(A) \tan(A) + \cos^2(A) \cot(A) + 2\sin(A)\cos(A) = \tan(A) + \cot(A)
\]

This completes the proof.

---

Do you want details on any particular step or further clarification?

Here are 5 related questions:
1. How can we derive \(\tan(A)\) and \(\cot(A)\) from the sine and cosine functions?
2. What are the common trigonometric identities used in simplifying expressions?
3. Can you prove the identity \(\sin^2(A) + \cos^2(A) = 1\)?
4. How do trigonometric identities help in simplifying complex expressions?
5. What is the relationship between \(\tan(A)\) and \(\cot(A)\)?

**Tip:** Whenever dealing with trigonometric identities, always try to express everything in terms of sine and cosine to simplify the expression.

Explore the step-by-step proof of the trigonometric identity sin²AtanA + cos²AcotA + 2sinAcosA = tanA + cotA. This problem utilizes key trigonometric identities and simplification techniques. Suitable for advanced high school students (Grades 11-12).

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