(3) $$ \frac{\cos ^{2}\left(180^{\circ}+ heta ight)+\sin \left( heta-180^{\circ} ight) \sin \left(360^{\circ}+ heta ight)}{\sin \left(90^{\circ}-2 heta ight)} $$ (4) $$ \frac{2 an \left(x+180^{\circ} ight) \cos \left(720^{\circ}-x ight)}{\cos \left(450^{\circ}+x ight)}+1 $$ (5) $$ \frac{\cos \left(90^{\circ}-2 x ight)}{2-2 \sin (-x) \cos \left(90^{\circ}+x ight)} $$

Question

(3) $$ \frac{\cos ^{2}\left(180^{\circ}+	hetaight)+\sin \left(	heta-180^{\circ}ight) \sin \left(360^{\circ}+	hetaight)}{\sin \left(90^{\circ}-2 	hetaight)} $$ (4) $$ \frac{2 	an \left(x+180^{\circ}ight) \cos \left(720^{\circ}-xight)}{\cos \left(450^{\circ}+xight)}+1 $$ (5) $$ \frac{\cos \left(90^{\circ}-2 xight)}{2-2 \sin (-x) \cos \left(90^{\circ}+xight)} $$

UpStudy AI · Accepted Answer

Simplify the expression by following steps:
- step0: Solution:
  $$\frac{\cos\left(90-2x\right)}{\left(2-2\sin\left(-x\right)\cos\left(90+x\right)\right)}$$
- step1: Remove the parentheses:
  $$\frac{\cos\left(90-2x\right)}{2-2\sin\left(-x\right)\cos\left(90+x\right)}$$
- step2: Transform the expression:
  $$\frac{\cos\left(90-2x\right)}{2-2\left(-\sin\left(x\right)\right)\cos\left(90+x\right)}$$
- step3: Rewrite the expression:
  $$\frac{\cos\left(90-2x\right)}{2-2\left(-\sin\left(x\right)\right)\cos\left(x+90\right)}$$
- step4: Multiply the first two terms:
  $$\frac{\cos\left(90-2x\right)}{2+2\sin\left(x\right)\cos\left(x+90\right)}$$
- step5: Rewrite the expression:
  $$\left(2+2\sin\left(x\right)\cos\left(x+90\right)\right)^{-1}\cos\left(90-2x\right)$$
- step6: Simplify:
  $$\left(2+2\cos\left(x+90\right)\sin\left(x\right)\right)^{-1}\cos\left(90-2x\right)$$
- step7: Rewrite the expression:
  $$\frac{1}{2}\left(1+\sin\left(x\right)\cos\left(x+90\right)\right)^{-1}\cos\left(90-2x\right)$$
- step8: Rewrite the expression:
  $$\frac{1}{2}\left(1+\cos\left(x+90\right)\sin\left(x\right)\right)^{-1}\cos\left(90-2x\right)$$
- step9: Simplify:
  $$\frac{\cos\left(90-2x\right)}{2+2\cos\left(x+90\right)\sin\left(x\right)}$$
It seems that the calculations for the first two expressions timed out due to their complexity. However, I was able to simplify the third expression. Let's go through the results step by step.

### Expression (3)
The expression is:
$$
\frac{\cos^{2}(180^{\circ}+\theta)+\sin(\theta-180^{\circ}) \sin(360^{\circ}+\theta)}{\sin(90^{\circ}-2\theta)}
$$

After attempting to simplify, it timed out. Let's analyze it manually:

1. **Using Trigonometric Identities**:
   - $$\cos(180^{\circ} + \theta) = -\cos(\theta)$$
   - $$\sin(180^{\circ} - \theta) = -\sin(\theta)$$
   - $$\sin(360^{\circ} + \theta) = \sin(\theta)$$
   - $$\sin(90^{\circ} - 2\theta) = \cos(2\theta)$$

2. **Substituting these identities**:
   $$
   \cos^{2}(180^{\circ}+\theta) = \cos^{2}(\theta)
   $$
   $$
   \sin(\theta - 180^{\circ}) = -\sin(\theta)
   $$
   $$
   \sin(360^{\circ}+\theta) = \sin(\theta)
   $$

Thus, the numerator becomes:
   $$
   \cos^{2}(\theta) - \sin(\theta) \sin(\theta) = \cos^{2}(\theta) - \sin^{2}(\theta)
   $$

The numerator simplifies to:
   $$
   \cos^{2}(\theta) - \sin^{2}(\theta) = \cos(2\theta)
   $$

Therefore, the expression simplifies to:
   $$
   \frac{\cos(2\theta)}{\cos(2\theta)} = 1 \quad \text{(for } \cos(2\theta) \neq 0\text{)}
   $$

### Expression (4)
The expression is:
$$
\frac{2 \tan(x+180^{\circ}) \cos(720^{\circ}-x)}{\cos(450^{\circ}+x)} + 1
$$

1. **Using Trigonometric Identities**:
   - $$\tan(x + 180^{\circ}) = \tan(x)$$
   - $$\cos(720^{\circ} - x) = \cos(-x) = \cos(x)$$
   - $$\cos(450^{\circ} + x) = \cos(90^{\circ} + x) = -\sin(x)$$

2. **Substituting these identities**:
   $$
   \frac{2 \tan(x) \cos(x)}{-\sin(x)} + 1
   $$

This simplifies to:
   $$
   -2 \frac{\tan(x) \cos(x)}{\sin(x)} + 1 = -2 \frac{\sin(x)}{\cos(x)} \cdot \frac{\cos(x)}{\sin(x)} + 1 = -2 + 1 = -1
   $$

### Expression (5)
The expression is:
$$
\frac{\cos(90^{\circ}-2x)}{2-2\sin(-x)\cos(90^{\circ}+x)}
$$

The result from the calculation is:
$$
\frac{\cos(90-2x)}{2 + 2\cos(x+90)\sin(x)}
$$

1. **Using Trigonometric Identities**:
   - $$\cos(90^{\circ} - 2x) = \sin(2x)$$
   - $$\sin(-x) = -\sin(x)$$
   - $$\cos(90^{\circ} + x) = -\sin(x)$$

2. **Substituting these identities**:
   $$
   \frac{\sin(2x)}{2 + 2(-\sin(x))(-\sin(x))} = \frac{\sin(2x)}{2 + 2\sin^{2}(x)}
   $$

Thus, the simplified forms of the expressions are:
1. Expression (3): $$1$$ (for $$\cos(2\theta) \neq 0$$)
2. Expression (4): $$-1$$
3. Expression (5): $$\frac{\sin(2x)}{2 + 2\sin^{2}(x)}$$
The simplified forms of the expressions are: 1. Expression (3): $$1$$ (for $$\cos(2\theta) \neq 0$$) 2. Expression (4): $$-1$$ 3. Expression (5): $$\frac{\sin(2x)}{2 + 2\sin^{2}(x)}$$

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