Math Snap

STEP 1

1. The given equation involves trigonometric identities.
2. We will use trigonometric identities to simplify and prove the equation.
3. The goal is to manipulate the left-hand side to match the right-hand side, $\cos 2\theta$.

STEP 2

1. Simplify the fraction inside the brackets.
2. Simplify the overall expression.
3. Prove the equation by matching both sides.

STEP 3

Start by simplifying the fraction inside the brackets. Notice that $\cos 8\theta$ and $\cos 4\theta$ can be expressed using double angle identities. Use the identity for $\cos 2A = 2\cos^2 A - 1$.
For $\cos 8\theta$, use:
$$\cos 8\theta = 2\cos^2 4\theta - 1$$ Substitute this into the fraction:
$$\frac{\cos 8\theta - 1}{\cos 4\theta - 1} = \frac{(2\cos^2 4\theta - 1) - 1}{\cos 4\theta - 1}$$ $$= \frac{2\cos^2 4\theta - 2}{\cos 4\theta - 1}$$ $$= \frac{2(\cos^2 4\theta - 1)}{\cos 4\theta - 1}$$

STEP 4

Notice that $\cos^2 4\theta - 1$ can be factored using the identity $\cos^2 A - 1 = (\cos A - 1)(\cos A + 1)$:
$$\cos^2 4\theta - 1 = (\cos 4\theta - 1)(\cos 4\theta + 1)$$ Substitute back into the fraction:
$$\frac{2(\cos 4\theta - 1)(\cos 4\theta + 1)}{\cos 4\theta - 1}$$ Cancel out $(\cos 4\theta - 1)$ from the numerator and denominator:
$$= 2(\cos 4\theta + 1)$$

STEP 5

Substitute the simplified expression back into the original equation:
$$\frac{\sec 2\theta}{4} \times 2(\cos 4\theta + 1) = \cos 2\theta$$ Simplify the expression:
$$\frac{\sec 2\theta}{4} \times 2\cos 4\theta + \frac{\sec 2\theta}{4} \times 2 = \cos 2\theta$$ $$\frac{\sec 2\theta \cos 4\theta}{2} + \frac{\sec 2\theta}{2} = \cos 2\theta$$

STEP 6

Recall that $\sec 2\theta = \frac{1}{\cos 2\theta}$. Substitute this into the equation:
$$\frac{\cos 4\theta}{2\cos 2\theta} + \frac{1}{2\cos 2\theta} = \cos 2\theta$$ Combine the terms:
$$\frac{\cos 4\theta + 1}{2\cos 2\theta} = \cos 2\theta$$

Multiply both sides by $2\cos 2\theta$ to eliminate the fraction:
$$\cos 4\theta + 1 = 2\cos^2 2\theta$$ Use the identity $\cos 4\theta = 2\cos^2 2\theta - 1$:
$$2\cos^2 2\theta - 1 + 1 = 2\cos^2 2\theta$$ Both sides are equal, proving the equation.
The equation is verified as true.