If tanA + cotA = 2, then find the value of tan²A + cot²A.

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Given: tanA + cotA = 2

We need to find tan²A + cot²A.

Step 1: Square both sides of the given equation:
(tanA + cotA)² = 2²

Expanding the left side:
tan²A + 2(tanA)(cotA) + cot²A = 4

Step 2: Recall that tanA * cotA = 1 (since cotA = 1/tanA)

So, the equation becomes:
tan²A + 2(1) + cot²A = 4

Which simplifies to:
tan²A + cot²A + 2 = 4

Step 3: Subtract 2 from both sides:
tan²A + cot²A = 4 - 2 = 2

Wait, this contradicts the options given. Let's re-check the calculation.

Re-examining the step:
(tanA + cotA)² = tan²A + 2tanA*cotA + cot²A = 4

Since tanA*cotA = 1,

So, tan²A + cot²A + 2 = 4

Therefore, tan²A + cot²A = 4 - 2 = 2

But 2 is one of the options, so the correct answer is 2.

Double-checking the options: 2 is option 4.

Hence, the value of tan²A + cot²A is 2.

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