Find the value of:

$$
\frac{4\cos(90^\circ - heta)\sin^3(90^\circ + heta) - 4\sin(90^\circ + heta)\cos^3(90^\circ - heta)}{\cos\left(\frac{180^\circ + 8 heta}{2} ight)}
$$

Question

Find the value of:

$$
\frac{4\cos(90^\circ - 	heta)\sin^3(90^\circ + 	heta) - 4\sin(90^\circ + 	heta)\cos^3(90^\circ - 	heta)}{\cos\left(\frac{180^\circ + 8	heta}{2}ight)}
$$

Accepted Answer

1. **Numerator Simplification:**
   - $$ \cos(90^\circ - 	heta) = \sin	heta $$
   - $$ \sin(90^\circ + 	heta) = \cos	heta $$
   - Substitute and expand:
     $$
     4\sin	heta \cdot (\cos	heta)^3 - 4\cos	heta \cdot (\sin	heta)^3
     = 4\sin	heta\cos^3	heta - 4\cos	heta\sin^3	heta
     $$
   - Factor out $$4$$:
     $$
     4\left(\sin	heta\cos^3	heta - \cos	heta\sin^3	hetaight)
     = 4\sin	heta\cos	heta(\cos^2	heta - \sin^2	heta)
     $$
   - $$ \cos^2	heta - \sin^2	heta = \cos(2	heta) $$, $$ \sin	heta\cos	heta = \frac{1}{2}\sin(2	heta) $$:
     $$
     4 \cdot \frac{1}{2} \sin(2	heta) \cos(2	heta) = 2\sin(2	heta)\cos(2	heta) = \sin(4	heta)
     $$

2. **Denominator Simplification:**
   - $$ \cos\left(\frac{180^\circ + 8	heta}{2}ight) = \cos(90^\circ + 4	heta) = -\sin(4	heta) $$

3. **Combine:**
   $$
   \frac{\sin(4	heta)}{-\sin(4	heta)} = -1
   $$

### Final Answer:
$$
\boxed{-1}
$$

This matches the code simplification and step-by-step trigonometric identities.

Answer

Answer

Answer