1) Simplification (trigonometric ratios of allied angles)?

Simplify :
i) [cos^2 (2x180-A). tan^2 (180+A). sec^2 (180-A)] / [ sin^2 (3x180+ A). cosec^2 (-A). cot^2 (90+A)]

ii)[ cos(90+A). sec(270+A). sin(180-A) ] / [ cosec (-A). cos (270-A). tan (180+A) ]
Find values of x and y lying between 0 degree and 360 degree?

2) find values of x and y lying between 0 degree and 360 degree
i) sin 2x = 0. 6428
ii) cos (1/2y+71 degree) = -0 . 3420

cos^2(360-A) = cos^2(A)

tan^2(180+A) = tan^2(A)
sec^2(180-A) = sec^2(A)
sin^2(540+A) = sin^2(A)
csc^2(-A) = csc^2(A)
cot^2(90+A) = tan^2(A)
so, using those simplifications, we have

cos^2 * tan^2 * sec^2 = tan^2
sin^2 * csc^2 * tan^2 = tan^2
and the fraction is just 1

How far do you get with the others?

To simplify the given trigonometric expressions and find the values of x and y, we can follow these steps:

1) Simplification (trigonometric ratios of allied angles):

i) [cos^2 (2x180-A). tan^2 (180+A). sec^2 (180-A)] / [ sin^2 (3x180+ A). cosec^2 (-A). cot^2 (90+A)]

To simplify this expression, we can start by using the trigonometric identities to rewrite the trigonometric ratios of allied angles:
- cos(2x180 - A) = cos(360 - A) = cos(A) (as cos(360 - theta) = cos(theta))
- tan(180 + A) = -tan(A) (as tan(180 + theta) = -tan(theta))
- sec(180 - A) = -sec(A) (as sec(180 - theta) = -sec(theta))
- sin(3x180 + A) = -sin(A) (as sin(3x180 + theta) = -sin(theta))
- cosec(-A) = -cosec(A) (as cosec(-theta) = -cosec(theta))
- cot(90 + A) = -tan(A) (as cot(90 + theta) = 1 / tan(theta))

Using these identities, the given expression becomes:
[i.e., (cos^2 A * (-tan^2 A) * (-sec^2 A))] / [(-sin^2 A) * (-cosec^2 A) * (-tan^2 A)]

Simplifying the expression further, we can cancel out the negative signs, and the expression becomes:
[cos^2 A * tan^2 A * sec^2 A] / [sin^2 A * cosec^2 A * tan^2 A]

Now we can simplify by canceling out the repetitive trigonometric ratios:
[cos^2 A * sec^2 A] / [sin^2 A * cosec^2 A]
[cos^2 A * (1/cos^2 A)] / [(1/sin^2 A) * (1/sin^2 A)]
1 / sin^2 A

Therefore, the simplified expression is 1 / sin^2 A.

ii) [cos(90+A). sec(270+A). sin(180-A)] / [cosec(-A). cos(270-A). tan(180+A)]

Similar to the first part, we can use the trigonometric identities to rewrite the trigonometric ratios of allied angles:
- cos(90 + A) = -sin(A) (as cos(90 + theta) = sin(theta))
- sec(270 + A) = -cosec(A) (as sec(270 + theta) = -cosec(theta))
- sin(180 - A) = sin(A) (as sin(180 - theta) = sin(theta))
- cosec(-A) = -cosec(A) (as cosec(-theta) = -cosec(theta))
- cos(270 - A) = -sin(A) (as cos(270 - theta) = -sin(theta))
- tan(180 + A) = -tan(A) (as tan(180 + theta) = -tan(theta))

Using these identities, the given expression becomes:
[(-sin(A)) * (-cosec(A)) * sin(A)] / [-cosec(A) * (-sin(A)) * (-tan(A))]

Simplifying the expression further, we can cancel out the negative signs, and the expression becomes:
[sin(A) * cosec(A) * sin(A)] / [cosec(A) * sin(A) * tan(A)]

Now we can simplify by canceling out the repetitive trigonometric ratios:
[sin^2 A] / [tan(A)]

Therefore, the simplified expression is sin^2 A / tan(A).

To find the values of x and y that lie between 0 degrees and 360 degrees, we need additional information or equations attached to the given trigonometric expressions. Without any additional equations or information, we cannot determine the values of x and y.