Determine the value of sum tan pi/4+sin a=?

For the beginning, we'll substitute the function tan
pi/4 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 a
+ 1 = sin a + sin pi/2 

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

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

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

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

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

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

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

tan pi/4+sin a = [sin(a/2) +
cos(a/2)]^2