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

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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

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