# Calculate the expressions E(x)= 1+ sinx + cosx E(x)= sinx - cosx E(x)= 1 - cosx

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For the third expression, we'll transform the difference of 2 trigonometric function, with the same name, into a product.

We'll write the expression E(x)= 1 - cosx as a difference of 2 function with the same name, in this case, "cosine".

We'll substitute the value 1 = cos 0, so the

E(x)=1-cos x <=> E(x)=cos 0-cos x.

We'll transform, the difference into a product:

E(x)=2sin[(0+x)/2]*sin[(0-x)/2]=2sin(x/2)*sin(-x/2),

But sin(-a)= -sin a, so E(x)= -2sin(x/2)*sin(x/2),

**E(x) = -2sin(x/2)]^2**

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In order to solve the expression E(x)=sin x-cos x, we'll try to write the value cos x=sin [(pi/2)-x]. So,

E(x)=sin x-cos x

E(x)=sin x-sin [(pi/2)-x]

E(x)=2cos{[x+(pi/2)-x]/2}sin{[x-(pi/2)+x]/2},

We'll eliminate like terms

E(x)=2cos(pi/4)sin{[(2x/2)-(pi/4)]}

E(x)=2*(sqrt2)/2*sin{[(2x/2)-(pi/4)]}

**E(x)=(sqrt2)*sin{[(2x/2)-(pi/4)]}**

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Now, we'll solve the first expression:

**E(x)= 1+ sinx + cosx**

We'll write the expression 1 + cosx as a sum of 2 function with the same name, in this case, "cosine".

We'll substitute the value 1 = cos 0, so the

1+cos x <=> cos 0+cos x.

We'll transform, the difference into a product:

2cos[(0+x)/2]*cos[(0-x)/2]=2cos(x/2)*cos(-x/2),

But cos(-a)= cos a, so 2cos(x/2)*cos(-x/2)=2cos(x/2)*cos(x/2),

**2cos(x/2)*cos(-x/2)=2cos(x/2)]^2**

So, the expression will become:

E(x)= 2cos(x/2)]^2+ sinx

We'll write sin x = sin 2(x/2) = 2sin(x/2)*cos(x/2)

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

We'll factorize by 2cos(x/2):

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

We'll write cos (x/2) = sin [(pi/2)-(x/2)]

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

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

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

**E(x)=2sqrt2*cos(x/2)*cos(pi/4 - x/2)**

E(x)= 1+ sinx + cosx

E(x)= sinx - cosx

E(x)= 1 - cosx.

From the 1st 2 equations, we get:

1+sinx+cosx = sinx-cosx

1 = -2cosx.

Therefore cos x = -1/2.

So sinx = sqrt(1-cos^2x) = sqrt (1-(1/2)^2) = +or -sqrt3/4.

Therefore from the last equation E(x) = 1-cosx = 1- (-1/2) = 3/2.