Explanation:

Let’s break it down step by step:

- We’re given: cosec ? + cosec²? = 1.

- Set cosec ? = x. So, x + x² = 1 ? x² + x - 1 = 0.

- Using the quadratic formula: x = [-1 ± v(1 + 4)] / 2 = [-1 ± v5]/2.

- But cosec ? (which is 1/sin ?) should be > 1 or < -1 (since |sin ?| = 1).

- Only x = ( -1 + v5)/2 ˜ 0.618, and x = ( -1 - v5)/2 ˜ -1.618.

- Only x = -1.618 is valid for |x| > 1. So, cosec ? = -1.618.

- sin ? = 1/cosec ? ˜ -0.618.

- That means ? is an angle with negative sine, likely in the third or fourth quadrant.

- Now, cot²? = cosec²? - 1.

- cosec²? = x² = [(-1-v5)/2]² = (1+2v5+5)/4 = (6 + 2v5)/4 = (3 + v5)/2.

- cot²? = (3 + v5)/2 - 1 = (3 + v5 - 2)/2 = (1 + v5)/2.

Let’s call cot²? = y = (1 + v5)/2.

Now, we want to find:

cot¹²? - 3 cot¹°? + 3 cot8? - cot6?

or

y6 - 3 y5 + 3 y4 - y³

Let’s factor this. Notice:

(y³ - 1)³ = y? - 3y6 + 3y³ - 1

But our expression is a different power set, so let’s calculate numerically:

- y ˜ (1 + 2.236)/2 ˜ 1.618.

- y³ ˜ (1.618)³ = ~4.236

- y4 ˜ y³ * y = 4.236 * 1.618 ˜ 6.854

- y5 ˜ y4 * y = 6.854 * 1.618 ˜ 11.089

- y6 ˜ y5 * y = 11.089 * 1.618 ˜ 17.999

Plug in the values:

= 17.999 - 3*11.089 + 3*6.854 - 4.236

= 17.999 - 33.267 + 20.562 - 4.236

= (17.999 + 20.562) - (33.267 + 4.236)

= 38.561 - 37.503

= 1.058 ˜ 1

Now, look at the options:

- Option:1, -2

- Option:2, -1

- Option:3, 0

- Option:4, 1

So, Option 4, 1 is correct.

Each statement and option:

- We solved the quadratic for cosec ?, then found cot²?, plugged in values, and calculated the final expression directly.

- The only answer matching the calculation is 1.

- Options -2, -1, and 0 are not supported by the computation.

- The correct answer is option 4: 1.