Math Snap

STEP 1

1. We are given several trigonometric expressions involving cosine and need to determine which are equivalent.
2. Equivalent expressions will yield the same value for any angle $\theta$.
3. We will use trigonometric identities and algebraic simplification to compare the expressions.

STEP 2

1. Analyze each expression individually.
2. Simplify or rewrite expressions using trigonometric identities.
3. Compare simplified expressions to determine equivalence.

STEP 3

Analyze the expression $\cos^2 \theta$:
This expression represents $(\cos \theta)^2$, which is the square of the cosine of $\theta$.

STEP 4

Analyze the expression $\cos 2\theta$:
This expression represents the cosine of the angle $2\theta$. Using the double angle identity, we have:
$$\cos 2\theta = \cos^2 \theta - \sin^2 \theta = 2\cos^2 \theta - 1$$

STEP 5

Analyze the expression $\cos(\theta)^2$:
This expression is equivalent to $\cos^2 \theta$ because it represents the square of the cosine of $\theta$.

STEP 6

Analyze the expression $\cos \theta \cdot \cos \theta$:
This expression is equivalent to $(\cos \theta)^2$, which is $\cos^2 \theta$.

STEP 7

Analyze the expression $\cos(\theta^2)$:
This expression represents the cosine of the square of $\theta$, which is different from the other expressions as it involves $\theta^2$ inside the cosine function.

Compare the expressions:
- $\cos^2 \theta$, $\cos(\theta)^2$, and $\cos \theta \cdot \cos \theta$ are all equivalent because they represent the square of the cosine of $\theta$.
- $\cos 2\theta$ is not equivalent to the above because it involves a double angle identity.
- $\cos(\theta^2)$ is not equivalent to any of the others because it involves the cosine of $\theta^2$.
The equivalent expressions are $\cos^2 \theta$, $\cos(\theta)^2$, and $\cos \theta \cdot \cos \theta$.