(1) $$ \frac{\cos \left(90^{\circ}+ heta ight)}{\sin (- heta)} $$ $$ \begin{array}{ll} ext { (3) } \frac{\cos \left(90^{\circ}-\alpha ight)}{\sin \left(180^{\circ}-\alpha ight)} & ext { (2) } \frac{\sin \left(90^{\circ}+x ight)}{\sin \left(90^{\circ}-x ight)} \ ext { (4) } \frac{\sin \left(360^{\circ}- heta ight)+\cos \left(90^{\circ}+ heta ight)}{\sin (- heta)}\end{array} $$

Question

(1) $$ \frac{\cos \left(90^{\circ}+	hetaight)}{\sin (-	heta)} $$ $$ \begin{array}{ll}	ext { (3) } \frac{\cos \left(90^{\circ}-\alphaight)}{\sin \left(180^{\circ}-\alphaight)} & 	ext { (2) } \frac{\sin \left(90^{\circ}+xight)}{\sin \left(90^{\circ}-xight)} \ 	ext { (4) } \frac{\sin \left(360^{\circ}-	hetaight)+\cos \left(90^{\circ}+	hetaight)}{\sin (-	heta)}\end{array} $$

UpStudy AI · Accepted Answer

Simplify the expression by following steps:
- step0: Solution:
  $$\frac{\cos\left(90+\theta \right)}{\sin\left(-\theta \right)}$$
- step1: Rewrite the expression:
  $$\frac{\cos\left(\theta +90\right)}{\sin\left(-\theta \right)}$$
- step2: Transform the expression:
  $$\frac{\cos\left(\theta +90\right)}{-\sin\left(\theta \right)}$$
- step3: Rewrite the fraction:
  $$-\frac{\cos\left(\theta +90\right)}{\sin\left(\theta \right)}$$
- step4: Rewrite the expression:
  $$-\cos\left(\theta +90\right)\csc\left(\theta \right)$$
Calculate or simplify the expression $$ \cos(90 - alpha) / \sin(180 - alpha) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{\cos\left(90-alpha\right)}{\sin\left(180-alpha\right)}$$
- step1: Multiply the terms:
  $$\frac{\cos\left(90-a^{2}lph\right)}{\sin\left(180-alpha\right)}$$
- step2: Multiply the terms:
  $$\frac{\cos\left(90-a^{2}lph\right)}{\sin\left(180-a^{2}lph\right)}$$
- step3: Rewrite the expression:
  $$\sin^{-1}\left(180-a^{2}lph\right)\cos\left(90-a^{2}lph\right)$$
- step4: Rewrite the expression:
  $$\cos\left(90-a^{2}lph\right)\sin^{-1}\left(180-a^{2}lph\right)$$
- step5: Simplify:
  $$\cos\left(90-a^{2}lph\right)\csc\left(180-a^{2}lph\right)$$
- step6: Rewrite the expression:
  $$\csc\left(180-a^{2}lph\right)\cos\left(90-a^{2}lph\right)$$
Calculate or simplify the expression $$ \sin(90 + x) / \sin(90 - x) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{\sin\left(90+x\right)}{\sin\left(90-x\right)}$$
- step1: Rewrite the expression:
  $$\frac{\sin\left(x+90\right)}{\sin\left(90-x\right)}$$
- step2: Rewrite the expression:
  $$\sin^{-1}\left(90-x\right)\sin\left(x+90\right)$$
- step3: Rewrite the expression:
  $$\sin\left(x+90\right)\sin^{-1}\left(90-x\right)$$
- step4: Simplify:
  $$\sin\left(x+90\right)\csc\left(90-x\right)$$
- step5: Rewrite the expression:
  $$\csc\left(90-x\right)\sin\left(x+90\right)$$
Calculate or simplify the expression $$ (\sin(360 - \theta) + \cos(90 + \theta)) / \sin(-\theta) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{\left(\sin\left(360-\theta \right)+\cos\left(90+\theta \right)\right)}{\sin\left(-\theta \right)}$$
- step1: Remove the parentheses:
  $$\frac{\sin\left(360-\theta \right)+\cos\left(90+\theta \right)}{\sin\left(-\theta \right)}$$
- step2: Rewrite the expression:
  $$\frac{\sin\left(360-\theta \right)+\cos\left(\theta +90\right)}{\sin\left(-\theta \right)}$$
- step3: Transform the expression:
  $$\frac{\sin\left(360-\theta \right)+\cos\left(\theta +90\right)}{-\sin\left(\theta \right)}$$
- step4: Rewrite the fraction:
  $$-\frac{\sin\left(360-\theta \right)+\cos\left(\theta +90\right)}{\sin\left(\theta \right)}$$
- step5: Rewrite the expression:
  $$\left(-\sin\left(360-\theta \right)-\cos\left(\theta +90\right)\right)\sin^{-1}\left(\theta \right)$$
- step6: Rewrite the expression:
  $$-\sin^{-1}\left(\theta \right)\sin\left(360-\theta \right)-\sin^{-1}\left(\theta \right)\cos\left(\theta +90\right)$$
- step7: Calculate:
  $$-\csc\left(\theta \right)\sin\left(360-\theta \right)-\sin^{-1}\left(\theta \right)\cos\left(\theta +90\right)$$
- step8: Calculate:
  $$-\csc\left(\theta \right)\sin\left(360-\theta \right)-\csc\left(\theta \right)\cos\left(\theta +90\right)$$
- step9: Rewrite the expression:
  $$-\sin\left(360-\theta \right)\csc\left(\theta \right)-\cos\left(\theta +90\right)\csc\left(\theta \right)$$
Let's simplify each expression step by step:

1. **Expression (1)**: 
   $$
   \frac{\cos(90^{\circ} + \theta)}{\sin(-\theta)}
   $$
   Using the cosine addition formula, we know that:
   $$
   \cos(90^{\circ} + \theta) = -\sin(\theta)
   $$
   And since $$\sin(-\theta) = -\sin(\theta)$$, we can rewrite the expression as:
   $$
   \frac{-\sin(\theta)}{-\sin(\theta)} = 1
   $$
   Thus, the simplified form is:
   $$
   1
   $$

2. **Expression (2)**: 
   $$
   \frac{\cos(90^{\circ} - \alpha)}{\sin(180^{\circ} - \alpha)}
   $$
   We know that:
   $$
   \cos(90^{\circ} - \alpha) = \sin(\alpha)
   $$
   And $$\sin(180^{\circ} - \alpha) = \sin(\alpha)$$. Therefore, the expression simplifies to:
   $$
   \frac{\sin(\alpha)}{\sin(\alpha)} = 1
   $$
   Thus, the simplified form is:
   $$
   1
   $$

3. **Expression (3)**: 
   $$
   \frac{\sin(90^{\circ} + x)}{\sin(90^{\circ} - x)}
   $$
   We know that:
   $$
   \sin(90^{\circ} + x) = \cos(x)
   $$
   And $$\sin(90^{\circ} - x) = \cos(x)$$. Therefore, the expression simplifies to:
   $$
   \frac{\cos(x)}{\cos(x)} = 1
   $$
   Thus, the simplified form is:
   $$
   1
   $$

4. **Expression (4)**: 
   $$
   \frac{\sin(360^{\circ} - \theta) + \cos(90^{\circ} + \theta)}{\sin(-\theta)}
   $$
   We know that:
   $$
   \sin(360^{\circ} - \theta) = -\sin(\theta)
   $$
   And $$\cos(90^{\circ} + \theta) = -\sin(\theta)$$. Therefore, the expression becomes:
   $$
   \frac{-\sin(\theta) - \sin(\theta)}{-\sin(\theta)} = \frac{-2\sin(\theta)}{-\sin(\theta)} = 2
   $$
   Thus, the simplified form is:
   $$
   2
   $$

In summary, the simplified results for each expression are:
1. $$1$$
2. $$1$$
3. $$1$$
4. $$2$$
The simplified results are: 1. 1 2. 1 3. 1 4. 2

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