Math Snap

STEP 1

1. We are asked to factor the expression $4 \sin^2 \theta - \cos^2 \theta$.
2. The expression involves trigonometric identities that can be used to simplify it.

STEP 2

1. Use trigonometric identities to rewrite the expression.
2. Factor the resulting expression.

STEP 3

Recall the Pythagorean identity:
$$\sin^2 \theta + \cos^2 \theta = 1$$ This identity can help us express $\sin^2 \theta$ in terms of $\cos^2 \theta$ or vice versa.

STEP 4

Rewrite $4 \sin^2 \theta - \cos^2 \theta$ using the identity. We can express $\sin^2 \theta$ as:
$$\sin^2 \theta = 1 - \cos^2 \theta$$ Substitute this into the expression:
$$4(1 - \cos^2 \theta) - \cos^2 \theta$$

STEP 5

Simplify the expression:
$$4 - 4\cos^2 \theta - \cos^2 \theta$$ Combine like terms:
$$4 - 5\cos^2 \theta$$

Factor the expression $4 - 5\cos^2 \theta$. Notice that it is already in a simplified form and does not factor further over the real numbers without using complex numbers or additional identities.
Thus, the expression $4 \sin^2 \theta - \cos^2 \theta$ simplifies to:
$$4 - 5\cos^2 \theta$$