December 21st at 7 in the morning i think <span />
Answer:
It was not my intention to post that answer, as it does not solve the question, but hope it helps somehow.
Step-by-step explanation:
![$\text{b)} \frac{\sin(a)}{\sin(a)-\cos(a)} - \frac{\cos(a)}{\cos(a)-\sin(a)} = \frac{1+\cot^2 (a)}{1-\cot^2 (a)} $](https://tex.z-dn.net/?f=%24%5Ctext%7Bb%29%7D%20%5Cfrac%7B%5Csin%28a%29%7D%7B%5Csin%28a%29-%5Ccos%28a%29%7D%20-%20%20%20%5Cfrac%7B%5Ccos%28a%29%7D%7B%5Ccos%28a%29-%5Csin%28a%29%7D%20%3D%20%5Cfrac%7B1%2B%5Ccot%5E2%20%28a%29%7D%7B1-%5Ccot%5E2%20%28a%29%7D%20%24)
You want to verify this identity.
![$\frac{\sin(a)(\cos(a)-\sin(a))}{(\sin(a)-\cos(a))(\cos(a)-\sin(a))} - \frac{\cos(a)(\sin(a)-\cos(a))}{(\sin(a)-\cos(a))(\cos(a)-\sin(a))} = \frac{1+\cot^2 (a)}{1-\cot^2 (a)} $](https://tex.z-dn.net/?f=%24%5Cfrac%7B%5Csin%28a%29%28%5Ccos%28a%29-%5Csin%28a%29%29%7D%7B%28%5Csin%28a%29-%5Ccos%28a%29%29%28%5Ccos%28a%29-%5Csin%28a%29%29%7D%20-%20%20%20%5Cfrac%7B%5Ccos%28a%29%28%5Csin%28a%29-%5Ccos%28a%29%29%7D%7B%28%5Csin%28a%29-%5Ccos%28a%29%29%28%5Ccos%28a%29-%5Csin%28a%29%29%7D%20%3D%20%5Cfrac%7B1%2B%5Ccot%5E2%20%28a%29%7D%7B1-%5Ccot%5E2%20%28a%29%7D%20%24)
The common denominator is
![(\sin(a)-\cos(a))(\cos(a)-\sin(a))= \boxed{2\cos (a)\sin(a)-\cos ^2(a)-\sin ^2(a)}](https://tex.z-dn.net/?f=%28%5Csin%28a%29-%5Ccos%28a%29%29%28%5Ccos%28a%29-%5Csin%28a%29%29%3D%20%5Cboxed%7B2%5Ccos%20%28a%29%5Csin%28a%29-%5Ccos%20%5E2%28a%29-%5Csin%20%5E2%28a%29%7D)
Solving the first and second numerator:
![\sin(a)(\cos(a)-\sin(a))=\sin(a)\cos(a)-\sin^2(a)](https://tex.z-dn.net/?f=%5Csin%28a%29%28%5Ccos%28a%29-%5Csin%28a%29%29%3D%5Csin%28a%29%5Ccos%28a%29-%5Csin%5E2%28a%29)
![\cos(a)(\sin(a)-\cos(a))= \cos(a)\sin(a)-\cos^2(a)](https://tex.z-dn.net/?f=%5Ccos%28a%29%28%5Csin%28a%29-%5Ccos%28a%29%29%3D%20%5Ccos%28a%29%5Csin%28a%29-%5Ccos%5E2%28a%29)
Now we have
![$\frac{ \sin(a)\cos(a)-\sin^2(a) -(\cos(a)\sin(a)-\cos^2(a))}{2\cos (a)\sin(a)-\cos ^2(a)-\sin ^2(a)}$](https://tex.z-dn.net/?f=%24%5Cfrac%7B%20%5Csin%28a%29%5Ccos%28a%29-%5Csin%5E2%28a%29%20-%28%5Ccos%28a%29%5Csin%28a%29-%5Ccos%5E2%28a%29%29%7D%7B2%5Ccos%20%28a%29%5Csin%28a%29-%5Ccos%20%5E2%28a%29-%5Csin%20%5E2%28a%29%7D%24)
![$\frac{ -\sin^2(a) +\cos^2(a)}{2\cos (a)\sin(a)-\cos ^2(a)-\sin ^2(a)}$](https://tex.z-dn.net/?f=%24%5Cfrac%7B%20-%5Csin%5E2%28a%29%20%2B%5Ccos%5E2%28a%29%7D%7B2%5Ccos%20%28a%29%5Csin%28a%29-%5Ccos%20%5E2%28a%29-%5Csin%20%5E2%28a%29%7D%24)
Once
![-\sin^2(a) +\cos^2(a) = \cos(2a)](https://tex.z-dn.net/?f=-%5Csin%5E2%28a%29%20%2B%5Ccos%5E2%28a%29%20%3D%20%5Ccos%282a%29)
![2\cos (a)\sin(a) = \sin(2a)](https://tex.z-dn.net/?f=2%5Ccos%20%28a%29%5Csin%28a%29%20%3D%20%5Csin%282a%29)
Also, consider the identity:
![\boxed{\sin^2(a)+\cos^2(a)=1}](https://tex.z-dn.net/?f=%5Cboxed%7B%5Csin%5E2%28a%29%2B%5Ccos%5E2%28a%29%3D1%7D)
![$\frac{ -\sin^2(a) +\cos^2(a)}{2\cos (a)\sin(a)-\cos ^2(a)-\sin ^2(a)}=\boxed{\frac{ \cos(2a)}{\sin(2a)-1}}$](https://tex.z-dn.net/?f=%24%5Cfrac%7B%20-%5Csin%5E2%28a%29%20%2B%5Ccos%5E2%28a%29%7D%7B2%5Ccos%20%28a%29%5Csin%28a%29-%5Ccos%20%5E2%28a%29-%5Csin%20%5E2%28a%29%7D%3D%5Cboxed%7B%5Cfrac%7B%20%5Ccos%282a%29%7D%7B%5Csin%282a%29-1%7D%7D%24)
That last claim is true.
144 seats are filled in the movie theater
The area of a circle is represented by the equation:
![A=\pi r^{2}](https://tex.z-dn.net/?f=%20A%3D%5Cpi%20r%5E%7B2%7D%20%20)
We know that the diameter is two times the radius: ![d=2r](https://tex.z-dn.net/?f=%20d%3D2r%20)
So if we know that the diameter of the circle is 24m, we can divide this by two in order to get the radius:
-->![r=12m](https://tex.z-dn.net/?f=%20r%3D12m%20)
So then we can plug this radius in to the equation for the area of a circle:
![A=\pi (12)^{2}](https://tex.z-dn.net/?f=%20A%3D%5Cpi%20%2812%29%5E%7B2%7D%20%20)
![A=144\pi](https://tex.z-dn.net/?f=%20A%3D144%5Cpi%20%20)
We are told to use 3.14 as pi, and not pi itself, so let's plug in 3.14 for pi:
![A=3.14(144)](https://tex.z-dn.net/?f=%20A%3D3.14%28144%29%20)
![A=452.16m^{2}](https://tex.z-dn.net/?f=%20A%3D452.16m%5E%7B2%7D%20%20)
Now we know that the area of this circle is 452.16 square meters.
Answer:
50 =c×d
Step-by-step explanation: