In a triangle ABC, right angled at B, if tan C = √3, then find the value of (sin²C + cos²C) / (1 + cot²C).

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Given: Triangle ABC is right angled at B.

We know tan C = √3.

Step 1: Recall the Pythagorean identity:
   sin²C + cos²C = 1.

Step 2: Recall the identity:
   1 + cot²C = csc²C.

Step 3: Substitute tan C = √3.
   cot C = 1 / tan C = 1 / √3 = √3 / 3.

Step 4: Calculate 1 + cot²C:
   1 + (√3 / 3)² = 1 + (3 / 9) = 1 + 1/3 = 4/3.

Step 5: Calculate the expression:
   (sin²C + cos²C) / (1 + cot²C) = 1 / (4/3) = 3/4.

Hence, the answer is 3/4.

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