To solve this problem, we can use the trigonometric identities:

cot²θ + 1 = csc²θ

cotθ = 1/tanθ

Using these identities, we can rewrite each term in the equation as follows:

cot² 60° = (1/tan 60°)² = (1/√3)² = 1/3

cot² 45° = (1/tan 45°)² = 1² = 1

cot² 30° = (1/tan 30°)² = (√3/3)² = 1/3

cosec² 60° = (1/sin 60°)² = (2/√3)² = 4/3

cosec² 45° = (1/sin 45°)² = 2² = 4

cosec² 30° = (1/sin 30°)² = (2/1)² = 4

Substituting these values into the equation, we get:

1/3 + 1 + 1/3 - 4/3 - 4 - 4 = -11

Therefore, cotan² 60° + cotan² 45° + cotan² 30° - cosec² 60° - cosec² 45° - cosec² 30° = -11.

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