Question 3 3.1. Simplify the fo 1.1. $$ \frac{\sin 135^{\circ} \cdot \cos 225^{\circ}+}{ an 315^{\circ}} $$ 1.2. $$ \frac{\sin \left(360^{\circ}-x ight) \cdot an ( }{\cos \left(180^{\circ}+x ight) an (1} $$

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Question 3 3.1. Simplify the fo 1.1. $$ \frac{\sin 135^{\circ} \cdot \cos 225^{\circ}+}{	an 315^{\circ}} $$ 1.2. $$ \frac{\sin \left(360^{\circ}-xight) \cdot 	an ( }{\cos \left(180^{\circ}+xight) 	an (1} $$

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Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\sin\left(135\right)\cos\left(225\right)}{\tan\left(315\right)}$$
- step1: Transform the expression:
  $$\frac{\sin\left(135\right)\cos\left(225\right)}{\frac{\sin\left(315\right)}{\cos\left(315\right)}}$$
- step2: Multiply by the reciprocal:
  $$\sin\left(135\right)\cos\left(225\right)\times \frac{\cos\left(315\right)}{\sin\left(315\right)}$$
- step3: Multiply the terms:
  $$\frac{\sin\left(135\right)\cos\left(225\right)\cos\left(315\right)}{\sin\left(315\right)}$$
- step4: Transform the expression:
  $$\sin\left(135\right)\cos\left(225\right)\cot\left(315\right)$$
Calculate or simplify the expression $$ \sin(360-x) * \tan(1) / (\cos(180+x) * \tan(1)) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{\sin\left(360-x\right)\tan\left(1\right)}{\left(\cos\left(180+x\right)\tan\left(1\right)\right)}$$
- step1: Remove the parentheses:
  $$\frac{\sin\left(360-x\right)\tan\left(1\right)}{\cos\left(180+x\right)\tan\left(1\right)}$$
- step2: Rewrite the expression:
  $$\frac{\sin\left(360-x\right)\tan\left(1\right)}{\cos\left(x+180\right)\tan\left(1\right)}$$
- step3: Multiply the terms:
  $$\frac{\tan\left(1\right)\sin\left(360-x\right)}{\cos\left(x+180\right)\tan\left(1\right)}$$
- step4: Multiply the terms:
  $$\frac{\tan\left(1\right)\sin\left(360-x\right)}{\tan\left(1\right)\cos\left(x+180\right)}$$
- step5: Reduce the fraction:
  $$\frac{\sin\left(360-x\right)}{\cos\left(x+180\right)}$$
- step6: Rewrite the expression:
  $$\cos^{-1}\left(x+180\right)\sin\left(360-x\right)$$
- step7: Rewrite the expression:
  $$\sin\left(360-x\right)\cos^{-1}\left(x+180\right)$$
- step8: Simplify:
  $$\sin\left(360-x\right)\sec\left(x+180\right)$$
- step9: Rewrite the expression:
  $$\sec\left(x+180\right)\sin\left(360-x\right)$$
Let's simplify the given expressions step by step.

### 3.1. Simplify the following expressions:

#### 1.1. 
$$
\frac{\sin 135^{\circ} \cdot \cos 225^{\circ}}{\tan 315^{\circ}}
$$

**Step 1: Identify the trigonometric values.**
- $$\sin 135^{\circ} = \sin(180^{\circ} - 45^{\circ}) = \sin 45^{\circ} = \frac{\sqrt{2}}{2}$$
- $$\cos 225^{\circ} = \cos(180^{\circ} + 45^{\circ}) = -\cos 45^{\circ} = -\frac{\sqrt{2}}{2}$$
- $$\tan 315^{\circ} = \tan(360^{\circ} - 45^{\circ}) = -\tan 45^{\circ} = -1$$

**Step 2: Substitute these values into the expression.**
$$
\frac{\frac{\sqrt{2}}{2} \cdot \left(-\frac{\sqrt{2}}{2}\right)}{-1} = \frac{-\frac{2}{4}}{-1} = \frac{\frac{1}{2}}{1} = \frac{1}{2}
$$

**Final Result for 1.1:**
$$
\sin(135) \cos(225) \cot(315)
$$

#### 1.2. 
$$
\frac{\sin(360^{\circ}-x) \cdot \tan(1)}{\cos(180^{\circ}+x) \cdot \tan(1)}
$$

**Step 1: Identify the trigonometric values.**
- $$\sin(360^{\circ}-x) = -\sin(x)$$
- $$\cos(180^{\circ}+x) = -\cos(x)$$

**Step 2: Substitute these values into the expression.**
$$
\frac{-\sin(x) \cdot \tan(1)}{-\cos(x) \cdot \tan(1)} = \frac{\sin(x)}{\cos(x)} = \tan(x)
$$

**Final Result for 1.2:**
$$
\sec(x+180) \sin(360-x)
$$

### Summary of Results:
1.1: $$\sin(135) \cos(225) \cot(315)$$

1.2: $$\sec(x+180) \sin(360-x)$$
1.1: $$\sin(135) \cos(225) \cot(315) = \frac{1}{2}$$ 1.2: $$\sec(x+180) \sin(360-x) = \tan(x)$$

Question 3 3.1. Simplify the fo 1.1. \( \frac{\sin 135^{\circ} \cdot \cos 225^{\circ}+}{\tan 315^{\circ}} \) 1.2. \( \frac{\sin \left(360^{\circ}-x\right) \cdot \tan ( }{\cos \left(180^{\circ}+x\right) \tan (1} \)

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