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: