calculate the sum sin x+square root(1-sin^2x)

To calculate the expression we'll have to transform the
sum into a product. The terms of the su are not like trigonometric
functions. 

We'll re-write the second term: sqrt(1-sin^2x)
= sqrt (cos x)^2

sin x + sqrt(1-sin^2x) = sin x + sqrt (cos
x)^2

sin x + sqrt(1-sin^2x) = sin x + cos
x

We'll  express the function cosine,
depending on the function sine.

cos x= sin
(90-x)

The expression will
become:

 sin x + cos x = sinx + sin
(90-x)

Now we can transform the expression into a
product:

 sin x + cos x = 2 sin (x+90-x)/2*cos
(x-90+x)/2

 sin x + cos x = 2 sin 45*cos
[-(90-2x)/2]

 sin x + cos x =  2* (sqrt2/2)*cos
(45-x)

 sin x + cos x =  sqrt 2*(cos 45*cos x
+ sin 45*sin x)