Reduce the following to the sine or cosine of one angle:

(i) sin145*cos75 - cos145*sin75
(ii) cos35*cos15 - sin35*sin15

Use the formulae:

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

and

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

(1)The quantity = sin(145-75)
= sin 70
= cos 20
note that If a+b=90 degrees then
sin a = cos b
(2) The quantity = cos (35 + 15)
= cos 50
= sin 40

To reduce the given expressions to the sine or cosine of one angle, we can use the trigonometric identities mentioned.

For expression (i), we can rewrite it as:
sin(145)*cos(75) - cos(145)*sin(75)

Using the identity sin(a+b) = sin(a) cos(b) + cos(a) sin(b), we get:
sin(145-75) = sin(70)
This is equal to cos(20) because if a + b = 90 degrees, then sin(a) = cos(b).

Therefore, the reduced form of expression (i) is cos(20).

For expression (ii), we can rewrite it as:
cos(35)*cos(15) - sin(35)*sin(15)

Using the identity cos(a+b) = cos(a) cos(b) - sin(a) sin(b), we get:
cos(35+15) = cos(50)
This is equal to sin(40).

Therefore, the reduced form of expression (ii) is sin(40).

In summary:
(i) sin(145)*cos(75) - cos(145)*sin(75) reduces to cos(20).
(ii) cos(35)*cos(15) - sin(35)*sin(15) reduces to sin(40).