$$ \frac { 2 \sin heta + \cos heta heta - \frac { \sin heta } { 1 - \cos heta } } { 1 - \cos 2 heta } $$

Question

$$ \frac { 2 \sin 	heta + \cos 	heta 	heta - \frac { \sin 	heta } { 1 - \cos 	heta } } { 1 - \cos 2 	heta } $$

UpStudy AI · Accepted Answer

To simplify the expression

$$
\frac{2 \sin \theta + \cos \theta \theta - \frac{\sin \theta}{1 - \cos \theta}}{1 - \cos 2\theta},
$$

we start by simplifying the denominator. We know that

$$
\cos 2\theta = 1 - 2\sin^2 \theta,
$$

so

$$
1 - \cos 2\theta = 2\sin^2 \theta.
$$

Now, substituting this into the expression gives us:

$$
\frac{2 \sin \theta + \cos \theta \theta - \frac{\sin \theta}{1 - \cos \theta}}{2 \sin^2 \theta}.
$$

Next, we simplify the numerator. The term $$-\frac{\sin \theta}{1 - \cos \theta}$$ can be rewritten. We know that

$$
1 - \cos \theta = 2 \sin^2 \left(\frac{\theta}{2}\right),
$$

so

$$
-\frac{\sin \theta}{1 - \cos \theta} = -\frac{\sin \theta}{2 \sin^2 \left(\frac{\theta}{2}\right)}.
$$

However, we can also leave it as is for now. Thus, the numerator becomes:

$$
2 \sin \theta + \cos \theta \theta - \frac{\sin \theta}{1 - \cos \theta}.
$$

Now, we can combine the terms in the numerator:

$$
= 2 \sin \theta + \cos \theta \theta - \frac{\sin \theta}{1 - \cos \theta}.
$$

To combine $$2 \sin \theta$$ and $$-\frac{\sin \theta}{1 - \cos \theta}$$, we need a common denominator:

$$
= \frac{2 \sin \theta (1 - \cos \theta) - \sin \theta}{1 - \cos \theta} + \cos \theta \theta.
$$

This simplifies to:

$$
= \frac{(2 \sin \theta - \sin \theta + 2 \sin \theta \cos \theta)}{1 - \cos \theta} + \cos \theta \theta,
$$

which simplifies to:

$$
= \frac{\sin \theta (1 + 2 \cos \theta)}{1 - \cos \theta} + \cos \theta \theta.
$$

Now, substituting this back into our expression gives:

$$
\frac{\frac{\sin \theta (1 + 2 \cos \theta)}{1 - \cos \theta} + \cos \theta \theta}{2 \sin^2 \theta}.
$$

This is a complex expression, and further simplification may depend on specific values of $$\theta$$ or additional context.

Thus, the simplified form of the original expression is:

$$
\frac{\sin \theta (1 + 2 \cos \theta) + \cos \theta \theta (1 - \cos \theta)}{2 \sin^2 \theta (1 - \cos \theta)}.
$$

This is as simplified as we can get without specific values for $$\theta$$.
The simplified form of the expression is: $$ \frac{\sin \theta (1 + 2 \cos \theta) + \cos \theta \theta (1 - \cos \theta)}{2 \sin^2 \theta (1 - \cos \theta)}. $$

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