We have to verify if sin 75 = (sqrt 6 + sqrt 2)/4

We know that sin 30 = 1/2 and cos 30 = (sqrt 3)/2 , cos 45 = 1/sqrt 2 and sin 45 = 1/sqrt 2

sin 75

=> sin(45 + 30)

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

=> sin 45* cos 30 + cos 45*cos 30

=> [1/(sqrt 2)]*(sqrt 3/2) + [1/(sqrt 2)]*(1/2)

=> (sqrt 3)/(2*sqrt 2) + 1/(2*sqrt 2)

=> (1 + sqrt 3)/(2*sqrt 2)

=> sqrt 2 (1 + sqrt 3)/4

=> (sqrt 2 + sqrt 6)/4

**This verifies that sin 75 = (sqrt 2 + sqrt 6)/4**

We'll write 75 as a sum of 2 angles:

75 = 45 + 30

We'll apply sine function both sides:

sin(75) = sin(45+30)

We'll use the identity:

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

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

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

sin(45+30) = sqrt(2*3)/4 + sqrt2/4

sin(45+30) = (sqrt6 + sqrt2)/4

**We notice that we get the indicated result, therefore the identity sin(75) = (sqrt6 + sqrt2)/4 is verified.**