Angle Addition Identities
There are some key identities that allow decomposition of angle addition in the sine, cosine, and tangent functions.
![](img/angleAddition.svg)
In the diagram above, the length of the green line is `\sin(x+y)`.
The segment beneath the horizontal is equal in length to the right edge of the rectangle, featured in blue, with a length of `\cos(x)\sin(y)`.
The length of the upper segment is given by `\sin(x)\cos(y)`, as formed by the triangle in the upper-right.
Altogether,
`\boxed{\sin(x+y)=\sin(x)\cos(y)+\cos(x)\sin(y)}`
Next, the length of the blue segment left of the green vertical is `\cos(x+y)`, while the complete length of the line is given by `\cos(x)\cos(y)`.
The length of the segment on the right is `\sin(x)\sin(y)`, as shown by the triangle in the upper-right corner.
Therefore, the part of the blue line left of the green is equal to:
`\boxed{\cos(x+y)=\cos(x)\cos(y)-\sin(x)\sin(y)}`
Dividing the two formulas:
`\displaystyle\frac{\sin(x+y)}{\cos(x+y)}=\frac{\sin(x)\cos(y)+\cos(x)\sin(y)}{\cos(x)\cos(y)-\sin(x)\sin(y)}=\boxed{\tan(x+y)=\frac{\tan(x)+\tan(y)}{1-\tan(x)\tan(y)}}`
Proofs taken from
Khan Academy.