find without tables or calculator.

(1) sin^2 (22 1/2)- cos^2 (22 1/2)

(2) sin 60 cos 30 + sin 30 cos 60

(3) cos(90-y) = sin 56degree 47^i

(1) should remind you of the formula for cos 2x. It looks like -cos (2*22.5). You should know what cos 45 is.

(2) should remind you of the formula for sin (a + b)

(3) I have no idea what 47^i means, but cos (90-y) = sin y

To solve these trigonometric expressions without using tables or a calculator, we can make use of the trigonometric identities and special angles.

(1) sin^2(22 1/2) - cos^2(22 1/2):

First, let's recall the double-angle identities:
sin^2(a) = (1 - cos(2a)) / 2
cos^2(a) = (1 + cos(2a)) / 2

Now, substitute a = 22 1/2 in the identities:
sin^2(45/2) = (1 - cos(45)) / 2
cos^2(45/2) = (1 + cos(45)) / 2

Next, let's use the special angle values:
cos(45) = sin(45) = √2 / 2

Substitute these values into the identities:
sin^2(45/2) = (1 - √2/2) / 2
cos^2(45/2) = (1 + √2/2) / 2

Finally, calculate the expressions:
sin^2(45/2) - cos^2(45/2) = [(1 - √2/2) / 2] - [(1 + √2/2) / 2]
= (1 - √2/2 - 1 - √2/2) / 2
= (-2√2/2) / 2
= -√2/2

Therefore, sin^2(22 1/2) - cos^2(22 1/2) = -√2/2.

(2) sin 60 cos 30 + sin 30 cos 60:

Using the values for sin(60) = √3/2, sin(30) = 1/2, cos(30) = √3/2, and cos(60) = 1/2:
sin 60 cos 30 + sin 30 cos 60 = (√3/2) * (√3/2) + (1/2) * (1/2)
= 3/4 + 1/4
= 4/4
= 1

Therefore, sin 60 cos 30 + sin 30 cos 60 = 1.

(3) cos(90-y) = sin 56 degrees:

First, let's use the complementary angle identity for cosine:
cos(90-y) = sin(y)

Given that cos(90-y) = sin(56 degrees), we can equate the two expressions:
sin(y) = sin(56 degrees)

Since the sine function is periodic with a period of 360 degrees, we need to consider multiple possible values of y. Here are two possibilities:

Using the fact that sin(a) = sin(b) if and only if a = b + k * 360, where k is an integer, we set up the equation:

y = 56 + k * 360

One possibility:
y = 56 + 0 * 360
= 56 degrees

Another possibility:
y = 180 - 56 + k * 360 (to account for the complementary angle)
= 124 + k * 360

In conclusion, the solutions for cos(90-y) = sin(56 degrees) are y = 56 degrees and y = 124 + k * 360 degrees, where k is an integer.