How do you use the half angle formulas to simplify the expression #-sqrt((1-cos8x)/(1+cos8x))#?

Question

Grace Ashcraft · Accepted Answer

#-|tan (4x)|# See Socratic graph for both.

The expression is #-sqrt(sin^2(4x)/cos^2(4x))# #=-|tan (4x)|, using sqrt(a^2)=|a|, by convention.practice/definition. graph{-sqrt((1-cos (8x))/(1+cos(8x)) [-0.8765, 0.8765, -0.438, 0.4385]} graph{-|tan(4x)| [-0.8765, 0.8765, -0.438, 0.4385]}

Josiah Anderson · Answer

undefined To simplify the expression $-\sqrt{\frac{1-\cos(8x)}{1+\cos(8x)}}$ using the half-angle formulas, follow these steps:

1. Start by expressing $1 - \cos(8x)$ and $1 + \cos(8x)$ in terms of a half-angle formula.
2. Apply the half-angle formulas to simplify.
3. Combine terms and simplify further if possible.

Using the half-angle identity for cosine:

$$\cos^2(\frac{\theta}{2}) = \frac{1 + \cos(\theta)}{2}$$

We have:

$$\frac{1 - \cos(8x)}{1 + \cos(8x)} = \frac{1 - \cos(4 \cdot 2x)}{1 + \cos(4 \cdot 2x)}$$

Using the half-angle identity, where $\theta = 4x$:

$$\cos^2(2x) = \frac{1 + \cos(4x)}{2}$$

This gives us:

$$\frac{1 - \cos(8x)}{1 + \cos(8x)} = \frac{1 - 2\cos^2(4x)}{1 + 2\cos^2(4x)}$$

Now, we can rewrite the expression as:

$$-\sqrt{\frac{1 - 2\cos^2(4x)}{1 + 2\cos^2(4x)}}$$

Simplify further if required.