**Using
the figure below, we will prove the Sine and Cosine Addition formulas
for the two angles x and y.**

**Note
that are all right triangles.**

**The
sine
addition formula
states that**

**sin(x+y)
= cos(x)sin(y) + sin(x)cos(y).**

**Let's
see
if we can prove this is true using the above Figure 1.**

**sin(x+y)
= .**

**Since
EDCF is a square, then segment EF is congruent to segment DC so...**

**sin
(x+y) = .**

**and
, then**

**sin(x+y)
= .**

**Finally,
since , we are left with sin(x+y) = siny cosx + cosy sinx.**

**Therefore,
we have proved the sine addition formula does work!**

**Note
again that are all right triangles.**

**The
cosine
addition formula
states that**

**cos(x+y)
= cos(x)cos(y) - sin(x)sin(y).**

**cos(x+y)
= .**

**Again,
since EDCF is a square, then segment FC is congruent to segment
ED, so substiting in for this, we get**

**cos(x+y)
= .**

**Since
, we get**

**cos(x+y)
= .**

**And
since , we are left with**

**cos(x+y)
= cosycosx - sinysinx.**

**We
have proved how the sine and cosine addition formulas work!**