For the start, we'll substitute the ratio sin45/cos45 by the function tan 45.

Then, we'll substitute tan 45 by it's value 1.

We'll transform the sum into a product. For this reason, we'll have to express the value 1 as being the function sine of an angle, so that the terms of the sum to be 2 matching trigonometric functions.

1 = sin pi/2

sin x + 1 = sin x + sin pi/2

sin x + sin pi/2 = 2sin [(x+pi/2)/2]*cos[ (x-pi/2)/2]

sin x + sin pi/2 = 2 sin [(x/2 + pi/4)]*cos[ (x/2 - pi/4)]

sin [(x/2 + pi/4)] = sin (x/2)*cos pi/4 + sin (pi/4)*cos (x/2)

sin [(x/2 + pi/4)] = (sqrt2/2)*[sin(x/2) + cos(x/2)]

cos[ (x/2 - pi/4)] = (sqrt2/2)*[sin(x/2) + cos(x/2)]

sin x + sin pi/2 = 2*(2/4)[sin(x/2) + cos(x/2)]^2

sin x + sin pi/2 = [sin(x/2) + cos(x/2)]^2

tan 45+sin x = [sin(x/2) + cos(x/2)]^2