Math Snap

STEP 1

What is this asking?
We need to find an equivalent expression to $\frac{\cos(2\theta)}{\sin^2 \theta}$ using trigonometric identities.
Watch out!
Remember the trigonometric identities, especially the double-angle formulas for cosine!
Also, be mindful of the values of $\theta$ for which the given expression is undefined.

STEP 2

1. Simplify using the double-angle formula
2. Transform to single-angle trigonometric functions

STEP 3

We're starting with $\frac{\cos(2\theta)}{\sin^2 \theta}$.
There are a few double-angle formulas for $\cos(2\theta)$, but since we have a $\sin^2 \theta$ in the denominator, the most useful one is $\cos(2\theta) = 1 - 2\sin^2 \theta$.
This will allow us to simplify things nicely!

STEP 4

Let's substitute this into our original expression:
$$\frac{\cos(2\theta)}{\sin^2 \theta} = \frac{1 - 2\sin^2 \theta}{\sin^2 \theta}$$ Now, we can split the fraction into two parts:
$$\frac{1 - 2\sin^2 \theta}{\sin^2 \theta} = \frac{1}{\sin^2 \theta} - \frac{2\sin^2 \theta}{\sin^2 \theta}$$ Since $\frac{\sin^2 \theta}{\sin^2 \theta} = 1$, we have:
$$\frac{1}{\sin^2 \theta} - 2$$

STEP 5

Remember that $\frac{1}{\sin \theta} = \csc \theta$, so $\frac{1}{\sin^2 \theta} = \csc^2 \theta$.
Let's use this!

STEP 6

Substituting this into our expression, we get:
$$\csc^2 \theta - 2$$ Now, we look at the available options and notice that none of them match exactly.
However, we remember another useful identity: $1 + \cot^2 \theta = \csc^2 \theta$.

STEP 7

Let's rewrite $\csc^2 \theta$ as $1 + \cot^2 \theta$:
$$\csc^2 \theta - 2 = (1 + \cot^2 \theta) - 2 = \cot^2 \theta - 1$$ Still no exact match!
But we know another identity: $1 = \sin^2 \theta + \cos^2 \theta$.
Substituting this for 1 gives us:
$$\cot^2 \theta - (\sin^2 \theta + \cos^2 \theta) = \cot^2 \theta - \sin^2 \theta - \cos^2 \theta$$ We can also use the identity $\cot^2 \theta = \csc^2 \theta - 1$ to rewrite our expression as:
$$\csc^2 \theta - 1 - 2 = \csc^2 \theta - 3$$ And since $1 = \sin^2 \theta + \cos^2 \theta$, we can rewrite 3 as $3(\sin^2 \theta + \cos^2 \theta)$.
Let's go back to $\csc^2 \theta - 2$.
We know that $\csc^2 \theta = 1 + \cot^2 \theta$, so we can write:
$$\csc^2 \theta - 2 = (1 + \cot^2 \theta) - 2 = \cot^2 \theta - 1$$ Since $1 = \csc^2 \theta - \cot^2 \theta$, we can write $2 = 2\csc^2 \theta - 2\cot^2 \theta$.
Then,
$$\csc^2 \theta - 2 = \csc^2 \theta - (2\csc^2 \theta - 2\cot^2 \theta) = 2\cot^2 \theta - \csc^2 \theta$$

The equivalent expression is $2 \cot^2 \theta - \csc^2 \theta$.