erify the identity.

(tan(*x*) + cot(*x*))2 = sec2(*x*) + csc2(*x*)

Expand the product, and use the Reciprocal and Pythagorean Identities to simplify.

(tan(*x*) + cot(*x*))2

=

tan2(*x*) + 2 tan(*x*) cot(*x*) +

=

tan2(*x*) + 2 +

=

tan2(*x*) + 1 + cot2(*x*) +

=

To verify the identity \((\tan(x) + \cot(x))^2 = \sec^2(x) + \csc^2(x)\), let’s expand and simplify the left side, as you've started. We'll use the reciprocal and Pythagorean identities to simplify further.

### Step-by-Step Solution

**1. Expand the Left Side:**
\[
(\tan(x) + \cot(x))^2 = \tan^2(x) + 2 \tan(x) \cot(x) + \cot^2(x)
\]

**2. Simplify Using the Product Identity of \(\tan(x)\) and \(\cot(x)\):**
Since \(\tan(x) \cot(x) = 1\),
\[
\tan^2(x) + 2 \cdot 1 + \cot^2(x) = \tan^2(x) + 2 + \cot^2(x)
\]

**3. Use Pythagorean Identities:**
Recall that:
\[
\tan^2(x) + 1 = \sec^2(x) \quad \text{and} \quad \cot^2(x) + 1 = \csc^2(x)
\]

Therefore:
\[
\tan^2(x) + \cot^2(x) + 2 = (\tan^2(x) + 1) + (\cot^2(x) + 1) = \sec^2(x) + \csc^2(x)
\]

**Conclusion:**
\[
(\tan(x) + \cot(x))^2 = \sec^2(x) + \csc^2(x)
\]

This verifies the identity.

Would you like further details or have any questions?

### Related Questions
1. How can the reciprocal identities simplify trigonometric expressions?
2. What are some other identities that involve \(\sec(x)\) and \(\csc(x)\)?
3. Why does the identity \(\tan(x) \cot(x) = 1\) hold?
4. How are Pythagorean identities derived in trigonometry?
5. Can similar techniques verify identities with other trigonometric functions?

### Tip:
Memorizing key trigonometric identities, like Pythagorean and reciprocal identities, can make simplifying expressions much quicker.

Verify the identity. (tan(x) + cot(x))^2 = sec^2(x) + csc^2(x). Expand the product, and use the Reciprocal and Pythagorean Identities to simplify.

Step-by-step guide to verifying the identity (tan(x) + cot(x))^2 = sec^2(x) + csc^2(x). This problem explores key trigonometric identities, including reciprocal and Pythagorean identities, and is suitable for high school students in Grades 10-12.

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