Failure is not an option. Use the product-to-sum identities to rewrite the following expressions as a sum or differe b. $$ 2 \cos (2 \beta) \cos (2 \beta) $$

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To rewrite the expression $$ 2 \cos(2\beta) \cos(2\beta) $$ using product-to-sum identities, we can start by recognizing that the expression can be simplified as follows:

$$
2 \cos(2\beta) \cos(2\beta) = 2 \cos^2(2\beta)
$$

However, if we want to use the product-to-sum identities directly, we can use the identity for the product of cosines:

$$
2 \cos A \cos B = \cos(A + B) + \cos(A - B)
$$

In this case, we have $$ A = 2\beta $$ and $$ B = 2\beta $$. Applying the identity:

$$
2 \cos(2\beta) \cos(2\beta) = \cos(2\beta + 2\beta) + \cos(2\beta - 2\beta)
$$

This simplifies to:

$$
\cos(4\beta) + \cos(0)
$$

Since $$ \cos(0) = 1 $$, we can write:

$$
\cos(4\beta) + 1
$$

Thus, the expression $$ 2 \cos(2\beta) \cos(2\beta) $$ can be rewritten as:

$$
\cos(4\beta) + 1
$$
$$ 2 \cos(2\beta) \cos(2\beta) = \cos(4\beta) + 1 $$

Failure is not an option. Use the product-to-sum identities to rewrite the following expressions as a sum or differe b. \( 2 \cos (2 \beta) \cos (2 \beta) \)

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