We want to show that:

1. Use the addition formula for

We use the addition formula:

Let

Then:

2. Use the double-angle formulas for

We use the known identities:

Substitute these into the equation:

3. Simplify the expression

First term:

Second term:

Add both terms:

Group terms:

4. Conclusion

We have now shown that:

5. Bonus: Multiple-Angle Formulas

The identity we just proved for is a specific case of what’s known as multiple-angle formulas in trigonometry. These formulas express and in terms of powers of and .

For example:

See: Multiple-Angle Trigonometric Formulas