Calculating the Least Value of Expression:

To calculate the least value of the expression cot2x-tan2x/1 sin(5π/2-8x), we need to follow these steps:

1. Simplify the Expression
2. Find the Domain of the Expression
3. Find the Critical Points
4. Determine the Sign of the Expression
5. Find the Least Value

Simplifying the Expression:

Let us simplify the given expression as follows:

cot2x-tan2x/1 sin(5π/2-8x)

= cot2x - tan2x / sin3(π/2 - 8x)

= (cos2x/sin2x) - (sin2x/cos2x) / (cos(π/2 - 8x) / sin2(π/2 - 8x))

= [(cos4x - sin4x) / (sin2x cos2x)] / (sin2(π/2 - 8x) / cos(8x))

= [(cos4x - sin4x) / (sin2x cos2x)] * (cos(8x) / sin(8x))

= [(cos2x + sin2x)(cos2x - sin2x) / (sin2x cos2x)] * (cos(8x) / sin(8x))

= [(1/tan2x)(1 - tan2x) / (1 + tan2x)] * (cos(8x) / sin(8x))

= [(1 - tan2x) / tan2x] * (cos(8x) / sin(8x))

= [(1/tan2x) - 1] * (cos(8x) / sin(8x))

= (cot2x - 1) * (cos(8x) / sin(8x))

Domain of the Expression:

The expression cot2x-tan2x/1 sin(5π/2-8x) is defined for all values of x except x = nπ/2 where n is an integer.

Critical Points:

The critical points of the expression occur when cot2x - 1 = 0, i.e., when cot2x = 1. This happens when x = nπ/4 where n is an integer.

Sign of the Expression:

For x < nπ/4,="" cot2x="" /> 1 and sin(8x) < 0.="" hence,="" the="" expression="" is="" />

For nπ/4 < x="" />< (n+1)π/4,="" cot2x="" />< 1="" and="" sin(8x)="" /> 0. Hence, the expression is positive.

For x > (n+1)π/4, cot2x > 1 and sin(8x) > 0. Hence, the expression is negative.

Least Value:

The least value of the expression occurs at the critical points. Let us evaluate the expression at x = nπ/4.

(cot2x - 1) * (cos(8x) / sin(8x)) = (1 - 1) * (cos(8nπ/4) / sin(8nπ/4)) =