2.1 / U.e 2.8 to cxpand $$ \cos 2 \alpha $$

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1. We start with the standard double‐angle formula for cosine:
   
   $$
   \cos 2\alpha = \cos^2 \alpha - \sin^2 \alpha.
   $$

2. Using the Pythagorean identity $$ \sin^2 \alpha + \cos^2 \alpha = 1 $$, we can write an alternative form. For instance, express $$ \sin^2 \alpha $$ as:
   
   $$
   \sin^2 \alpha = 1 - \cos^2 \alpha.
   $$

3. Substitute this into the original expression:
   
   $$
   \cos 2\alpha = \cos^2 \alpha - (1 - \cos^2 \alpha) = 2\cos^2 \alpha - 1.
   $$

4. Similarly, expressing $$ \cos^2 \alpha $$ as $$ 1 - \sin^2 \alpha $$ yields:
   
   $$
   \cos 2\alpha = (1 - \sin^2 \alpha) - \sin^2 \alpha = 1 - 2\sin^2 \alpha.
   $$

Thus, the expansion of $$ \cos 2\alpha $$ can be written equivalently as:

- $$\cos 2\alpha = \cos^2 \alpha - \sin^2 \alpha,$$
- $$\cos 2\alpha = 2\cos^2 \alpha - 1,$$
- $$\cos 2\alpha = 1 - 2\sin^2 \alpha.$$
The cosine of double angle $$ \cos 2\alpha $$ can be expanded in three ways: 1. $$ \cos 2\alpha = \cos^2 \alpha - \sin^2 \alpha $$ 2. $$ \cos 2\alpha = 2\cos^2 \alpha - 1 $$ 3. $$ \cos 2\alpha = 1 - 2\sin^2 \alpha $$

2.1 / U.e 2.8 to cxpand \( \cos 2 \alpha \)

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