If cot θ = √11, then find the value of (cosec²θ - sec²θ) / (cosec²θ + sec²θ).

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Given cot θ = √11.

Step 1: Express cot θ in terms of sin θ and cos θ:
  cot θ = cos θ / sin θ = √11

Step 2: Let sin θ = x, then cos θ = √11 * x.

Step 3: Use the Pythagorean identity:
  sin²θ + cos²θ = 1
  x² + (√11 * x)² = 1
  x² + 11x² = 1
  12x² = 1
  x² = 1/12
  sin²θ = 1/12

Step 4: Calculate cosec²θ and sec²θ:
  cosec²θ = 1 / sin²θ = 1 / (1/12) = 12
  cos²θ = 11 * sin²θ = 11 * (1/12) = 11/12
  sec²θ = 1 / cos²θ = 1 / (11/12) = 12/11

Step 5: Calculate numerator and denominator:
  Numerator = cosec²θ - sec²θ = 12 - 12/11 = (132/11) - (12/11) = 120/11
  Denominator = cosec²θ + sec²θ = 12 + 12/11 = (132/11) + (12/11) = 144/11

Step 6: Calculate the expression:
  (cosec²θ - sec²θ) / (cosec²θ + sec²θ) = (120/11) / (144/11) = 120/144 = 5/6

Therefore, the value of the expression is 5/6.

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