Rewriting sin, cos, tan in terms of half tangent

### 10.2.19 Rewriting sin, cos, tan in terms of half tangent

The halftan command rewrites the trigonometric functions in terms of tan(x/2) using the identities:

sin(x) = |
2tan | ⎛ ⎜ ⎜ ⎝ |
x
2
| ⎞ ⎟ ⎟ ⎠
tan2 | ⎛ ⎜ ⎜ ⎝ |
x
2
| ⎞ ⎟ ⎟ ⎠ | +1
| , cos(x) = |
1−tan2 | ⎛ ⎜ ⎜ ⎝ |
x
2
| ⎞ ⎟ ⎟ ⎠
tan2 | ⎛ ⎜ ⎜ ⎝ |
x
2
| ⎞ ⎟ ⎟ ⎠ | +1
| , tan(x) = |
2tan | ⎛ ⎜ ⎜ ⎝ |
x
2
| ⎞ ⎟ ⎟ ⎠
1−tan2 | ⎛ ⎜ ⎜ ⎝ |
x
2
| ⎞ ⎟ ⎟ ⎠
| .

- halftan takes expr, an expression containing trigonometric functions.
- halftan(expr) returns expr with any trigonometric functions replaced by the appropriate expression of tan(x/2).

#### Examples

halftan(sin(2*x)/(1+cos(2*x)))

2tan | ⎛ ⎜ ⎜ ⎝ |
2
2
| x | ⎞ ⎟ ⎟ ⎠
⎛ ⎜ ⎜ ⎝ | tan2 | ⎛ ⎜ ⎜ ⎝ |
2
2
| x | ⎞ ⎟ ⎟ ⎠ | +1 | ⎞ ⎟ ⎟ ⎠ | ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ | 1+ |
1−tan2 | ⎛ ⎜ ⎜ ⎝ |
2
2
| x | ⎞ ⎟ ⎟ ⎠
tan2 | ⎛ ⎜ ⎜ ⎝ |
2
2
| x | ⎞ ⎟ ⎟ ⎠ | +1
| ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

normal(ans())

tanx |  |  |  |  |  |  |  |  |  |

halftan(sin(x)^2+cos(x)^2)

| ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ |
2 tan | ⎛ ⎜ ⎜ ⎝ |
x
2
| ⎞ ⎟ ⎟ ⎠
tan2 | ⎛ ⎜ ⎜ ⎝ |
x
2
| ⎞ ⎟ ⎟ ⎠ | +1
| ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ |
2
| + | ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ |
1−tan2 | ⎛ ⎜ ⎜ ⎝ |
x
2
| ⎞ ⎟ ⎟ ⎠
tan2 | ⎛ ⎜ ⎜ ⎝ |
x
2
| ⎞ ⎟ ⎟ ⎠ | +1
| ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ |
2

normal(ans())

1 |  |  |  |  |  |  |  |  |  |