To calculate sin 75, we'll write 75 as the sum of 2 angles:

75 = 30 + 45

We'll apply sine function both sides:

sin 75 = sin (30+45)

To calculate sin (30+45), we'll apply the formula:

sin (a+b) = sin a*cos b + sin b*cos a

We'll put a = 30 and b = 45

sin (30+45) = sin 30*cos 45 + sin 45*cos 30

We'll substitute sin 30; sin 45; cos 30; cos 45 by their values:

sin 30 = 1/2

cos 30 = sqrt3/2

sin 45 = cos 45 = sqrt2/2

sin (30+45) = (1/2)*(sqrt2/2) + (sqrt2/2)*(sqrt3/2)

We'll factorize by (sqrt2/2):

sin (30+45) = (sqrt2/2)[(1+sqrt3)/2]

sin (30+45) = sqrt2*(1+sqrt3)/4

sin 75 = sqrt2*(1+sqrt3)/4

To calculate cos22deg30min, we'll write the formula for teh half-angle:

cos (a/2) = sqrt [(1+cos a)/2]

We'll put a = 22deg30min

2a = 2*22deg30min = 22deg30min + 22deg30min = 44deg + 1deg = 45 degrees

cos22deg30min = sqrt [(1+cos 45)/2]

cos22deg30min = sqrt [(1 + sqrt2/2)/2]

cos22deg30min = sqrt(2+sqrt2)/2