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

The expression is

#=-|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]}

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

- Start by expressing (1 - \cos(8x)) and (1 + \cos(8x)) in terms of a half-angle formula.
- Apply the half-angle formulas to simplify.
- 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.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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