Double-Angle and Multiple-Angle Identities

Double-Angle Identities

The double-angle identities are a special case of the addition formulas.

To derive the double-angle formula for sine, we use the sine addition formula:

Putting $$ gives

that is,

Similarly, substituting $$ in the cosine addition formula

yields the double-angle identity for cosine:

or

Using the Pythagorean trigonometric identity, we can get two more variations for the cosine of a double angle:

Since $$ we have

Hence, the first variation is given by

If we substitute the identity $$ in the original double-angle formula for cosine, we get the second variation:

Thus,

To derive the double-angle formula for tangent, recall the tangent addition formula

Let $$ Then

that is,

The double-angle identity for cotangent has the form

Hence,

Double-Angle Identities in Terms of Tangent

The double angle identities can be expressed in terms of the tangent of the single angle. To prove this, note that

Assuming $$ we can divide both the numerator and denominator by $$ This yields

We got the following identity:

Now take the double-angle formula for cosine:

Divide the numerator and denominator by $$ again, provided $$ or $$ $$

So, we have

Finally, note that $$ Therefore,

where $$ $$

Triple-Angle Identities

Using double-angle identities and addition formulas, we can derive triple-angle identities for trigonometric functions.

Let's start with the sine function:

Thus,

The triple-angle formula for cosine can be proved in a similar manner:

Hence,

The triple-angle identity for tangent is given by

that is,

Let's also consider the triple-angle formula for cotangent:

or