Answer:
Ste
Random draws are often used to make a decision where no rational or fair basis exists for making a deterministic decision, or to make unpredictable moves.p-by-step explanation:
This isn't a question, but if you make one I will help you the best I can.
If you buy one photo op picture you can have as many people as you want. ( there is a limit of course but I don’t know what it is)
It depends on the person you’re taking the photo with but most of the time, yes you can
Answer:
See below.
Step-by-step explanation:
![\cos(20)-\sin(20)=\sqrt{2}\sin(25)](https://tex.z-dn.net/?f=%5Ccos%2820%29-%5Csin%2820%29%3D%5Csqrt%7B2%7D%5Csin%2825%29)
First, use the co-function identity:
![\sin(90-x)=\cos(x)](https://tex.z-dn.net/?f=%5Csin%2890-x%29%3D%5Ccos%28x%29)
We can turn the second term into cosine:
![\sin(20)=\sin(90-70)=\cos(70)](https://tex.z-dn.net/?f=%5Csin%2820%29%3D%5Csin%2890-70%29%3D%5Ccos%2870%29)
Substitute:
![\cos(20)-\cos(70)=\sqrt{2}\sin(25)](https://tex.z-dn.net/?f=%5Ccos%2820%29-%5Ccos%2870%29%3D%5Csqrt%7B2%7D%5Csin%2825%29)
Now, use the sum to product formulas. We will use the following:
![\cos(x)-\cos(y)=-2\sin(\frac{x+y}{2})\sin(\frac{x-y}{2})](https://tex.z-dn.net/?f=%5Ccos%28x%29-%5Ccos%28y%29%3D-2%5Csin%28%5Cfrac%7Bx%2By%7D%7B2%7D%29%5Csin%28%5Cfrac%7Bx-y%7D%7B2%7D%29)
Substitute:
![\cos(20)-\cos(70)=-2\sin(\frac{20+70}{2})\sin(\frac{20-70}{2})\\\cos(20)-\cos(70) =-2\sin(45)\sin(-25)\\\cos(20)-\cos(70)=-2(\frac{\sqrt{2}}{2})\sin(-25)\\ \cos(20)-\cos(70)=-\sqrt{2}\sin(-25)](https://tex.z-dn.net/?f=%5Ccos%2820%29-%5Ccos%2870%29%3D-2%5Csin%28%5Cfrac%7B20%2B70%7D%7B2%7D%29%5Csin%28%5Cfrac%7B20-70%7D%7B2%7D%29%5C%5C%5Ccos%2820%29-%5Ccos%2870%29%20%20%3D-2%5Csin%2845%29%5Csin%28-25%29%5C%5C%5Ccos%2820%29-%5Ccos%2870%29%3D-2%28%5Cfrac%7B%5Csqrt%7B2%7D%7D%7B2%7D%29%5Csin%28-25%29%5C%5C%20%5Ccos%2820%29-%5Ccos%2870%29%3D-%5Csqrt%7B2%7D%5Csin%28-25%29)
Use the even-odd identity:
![\sin(-x)=-\sin(x)](https://tex.z-dn.net/?f=%5Csin%28-x%29%3D-%5Csin%28x%29)
Therefore:
![\cos(20)-\cos(70)=-\sqrt{2}\sin(-25)\\\cos(20)-\cos(70)=-\sqrt{2}\cdot-\sin(25)\\\cos(20)-\cos(70)=\sqrt{2}\sin(25)](https://tex.z-dn.net/?f=%5Ccos%2820%29-%5Ccos%2870%29%3D-%5Csqrt%7B2%7D%5Csin%28-25%29%5C%5C%5Ccos%2820%29-%5Ccos%2870%29%3D-%5Csqrt%7B2%7D%5Ccdot-%5Csin%2825%29%5C%5C%5Ccos%2820%29-%5Ccos%2870%29%3D%5Csqrt%7B2%7D%5Csin%2825%29)
Replace the second term with the original term:
![\cos(20)-\sin(20)=\sqrt{2}\sin(25)](https://tex.z-dn.net/?f=%5Ccos%2820%29-%5Csin%2820%29%3D%5Csqrt%7B2%7D%5Csin%2825%29)
Proof complete.