**Tangent half-angle substitution** is a technique primarily utilised in integral calculus.

## The substitution

### Differential

First, let Differentiate using the chain rule

- .

Consider the Pythagorean identity

Therefore,

Isolate ,

### Sine

Consider the double angle identity,

Which may be written as,

Which in turn may be written as,

Recall that,

We can therefore rewrite the above as,

Therefore,

- .

### Cosine

Consider the double angle identity,

.

Rewriting,

Writing in terms of t,

- .

## Examples

### Example 1

Let,

- .

Using,

- .

We may rewrite the integral as,

- .
- .

From there,

- .

Use the substitution,

- .

Therefore,

Isolate ,

So,

### Example 2

Let

Multiplying through

Rearranging,

- .

First case,

Apply arctangent to both sides

Second case,

Again, apply arctangent to both sides,

So the two general solutions are,

Where is any integer.