Prove sin²AtanA+cos²AcotA+2sinAcosA=tanA+cotA

We are given the trigonometric identity 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 \(\tan(A)\) and \(\cot(A)\):**
   We know the basic trigonometric identities:
   \[
   \tan(A) = \frac{\sin(A)}{\cos(A)} \quad \text{and} \quad \cot(A) = \frac{\cos(A)}{\sin(A)}
   \]
   Using these, rewrite the given equation:

   \[
   \sin^2(A) \cdot \frac{\sin(A)}{\cos(A)} + \cos^2(A) \cdot \frac{\cos(A)}{\sin(A)} + 2 \sin(A) \cos(A)
   \]
   Simplify the terms:
   \[
   = \frac{\sin^3(A)}{\cos(A)} + \frac{\cos^3(A)}{\sin(A)} + 2 \sin(A) \cos(A)
   \]

2. **Combine the fractions:**
   We will now combine the two fractions:
   \[
   \frac{\sin^3(A)}{\cos(A)} + \frac{\cos^3(A)}{\sin(A)}
   \]
   Find a common denominator:
   \[
   = \frac{\sin^4(A) + \cos^4(A)}{\sin(A)\cos(A)}
   \]

   Now the expression becomes:
   \[
   \frac{\sin^4(A) + \cos^4(A)}{\sin(A)\cos(A)} + 2 \sin(A) \cos(A)
   \]

3. **Simplify the expression further:**
   Observe that \(2 \sin(A) \cos(A) = \sin(2A)\), but for now, let's not use this. We'll instead express the entire equation over the common denominator \(\sin(A)\cos(A)\):

   \[
   \frac{\sin^4(A) + \cos^4(A)}{\sin(A)\cos(A)} + \frac{2 \sin^2(A) \cos^2(A)}{\sin(A)\cos(A)}
   \]
   Combine the numerators:
   \[
   = \frac{\sin^4(A) + \cos^4(A) + 2 \sin^2(A) \cos^2(A)}{\sin(A)\cos(A)}
   \]

4. **Use the identity for the sum of powers:**
   Notice that:
   \[
   \sin^4(A) + \cos^4(A) + 2 \sin^2(A) \cos^2(A) = (\sin^2(A) + \cos^2(A))^2
   \]
   And since \(\sin^2(A) + \cos^2(A) = 1\), we get:
   \[
   (\sin^2(A) + \cos^2(A))^2 = 1^2 = 1
   \]
   Therefore, the expression simplifies to:
   \[
   \frac{1}{\sin(A) \cos(A)}
   \]

5. **Rewrite using \(\sin(A) \cos(A)\):**
   Recall that:
   \[
   \frac{1}{\sin(A) \cos(A)} = \tan(A) + \cot(A)
   \]

### Final Answer:
Thus, we have 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.

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Do you want further explanation on any step or more details?

Here are 5 related questions you might find helpful:
1. What are some other common trigonometric identities used in proofs?
2. How can you derive the identities for \(\tan(A)\) and \(\cot(A)\) from sine and cosine?
3. Why is the Pythagorean identity \(\sin^2(A) + \cos^2(A) = 1\) so useful in trigonometric proofs?
4. Can this identity be extended to different angle relationships (e.g., double angles)?
5. What is the geometric interpretation of \(\sin(A)\), \(\cos(A)\), and \(\tan(A)\) on the unit circle?

**Tip:** Memorizing key identities, like \(\tan(A) = \frac{\sin(A)}{\cos(A)}\) and the Pythagorean identity, will greatly simplify trigonometric proofs!

This page provides a detailed step-by-step proof for the trigonometric identity sin²AtanA + cos²AcotA + 2sinAcosA = tanA + cotA. Learn about key trigonometric identities and simplifications, suitable for students in Grades 10-12.

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