Answer:
![\frac{\cot x}{1+\csc x}=\frac{\csc x-1}{\cot x}](https://tex.z-dn.net/?f=%5Cfrac%7B%5Ccot%20x%7D%7B1%2B%5Ccsc%20x%7D%3D%5Cfrac%7B%5Ccsc%20x-1%7D%7B%5Ccot%20x%7D)
Step-by-step explanation:
We want to verify the identity:
![\frac{\cot x}{1+\csc x}=\frac{\csc x-1}{\cot x}](https://tex.z-dn.net/?f=%5Cfrac%7B%5Ccot%20x%7D%7B1%2B%5Ccsc%20x%7D%3D%5Cfrac%7B%5Ccsc%20x-1%7D%7B%5Ccot%20x%7D)
Let us take the LHS and simplify to get the LHS.
Express everything in terms of the cosine and sine function.
![\frac{\cot x}{1+\csc x}=\frac{\frac{\cos x}{\sin x} }{1+\frac{1}{\sin x} }](https://tex.z-dn.net/?f=%5Cfrac%7B%5Ccot%20x%7D%7B1%2B%5Ccsc%20x%7D%3D%5Cfrac%7B%5Cfrac%7B%5Ccos%20x%7D%7B%5Csin%20x%7D%20%7D%7B1%2B%5Cfrac%7B1%7D%7B%5Csin%20x%7D%20%7D)
Collect LCM
![\frac{\cot x}{1+\csc x}=\frac{\frac{\cos x}{\sin x} }{\frac{\sin x+1}{\sin x} }](https://tex.z-dn.net/?f=%5Cfrac%7B%5Ccot%20x%7D%7B1%2B%5Ccsc%20x%7D%3D%5Cfrac%7B%5Cfrac%7B%5Ccos%20x%7D%7B%5Csin%20x%7D%20%7D%7B%5Cfrac%7B%5Csin%20x%2B1%7D%7B%5Csin%20x%7D%20%7D)
We simplify the RHS to get:
![\frac{\cot x}{1+\csc x}=\frac{\cos x}{\sin x+1}](https://tex.z-dn.net/?f=%5Cfrac%7B%5Ccot%20x%7D%7B1%2B%5Ccsc%20x%7D%3D%5Cfrac%7B%5Ccos%20x%7D%7B%5Csin%20x%2B1%7D)
We rationalize to get:
![\frac{\cot x}{1+\csc x}=\frac{\cos x(\sin x-1)}{(\sin x+1)*(\sin x-1)}](https://tex.z-dn.net/?f=%5Cfrac%7B%5Ccot%20x%7D%7B1%2B%5Ccsc%20x%7D%3D%5Cfrac%7B%5Ccos%20x%28%5Csin%20x-1%29%7D%7B%28%5Csin%20x%2B1%29%2A%28%5Csin%20x-1%29%7D)
We expand to get:
![\frac{\cot x}{1+\csc x}=\frac{\cos x(\sin x-1)}{\sin^2 x-1}](https://tex.z-dn.net/?f=%5Cfrac%7B%5Ccot%20x%7D%7B1%2B%5Ccsc%20x%7D%3D%5Cfrac%7B%5Ccos%20x%28%5Csin%20x-1%29%7D%7B%5Csin%5E2%20x-1%7D)
Factor negative one in the denominator:
![\frac{\cot x}{1+\csc x}=\frac{\cos x(\sin x-1)}{-(1-\sin^2 x)}](https://tex.z-dn.net/?f=%5Cfrac%7B%5Ccot%20x%7D%7B1%2B%5Ccsc%20x%7D%3D%5Cfrac%7B%5Ccos%20x%28%5Csin%20x-1%29%7D%7B-%281-%5Csin%5E2%20x%29%7D)
Apply the Pythagoras Property to get:
![\frac{\cot x}{1+\csc x}=\frac{\cos x(\sin x-1)}{-\cos^2 x}](https://tex.z-dn.net/?f=%5Cfrac%7B%5Ccot%20x%7D%7B1%2B%5Ccsc%20x%7D%3D%5Cfrac%7B%5Ccos%20x%28%5Csin%20x-1%29%7D%7B-%5Ccos%5E2%20x%7D)
Simplify to get:
![\frac{\cot x}{1+\csc x}=\frac{-(\sin x-1)}{\cos x}](https://tex.z-dn.net/?f=%5Cfrac%7B%5Ccot%20x%7D%7B1%2B%5Ccsc%20x%7D%3D%5Cfrac%7B-%28%5Csin%20x-1%29%7D%7B%5Ccos%20x%7D)
Or
![\frac{\cot x}{1+\csc x}=\frac{1-\sin x}{\cos x}](https://tex.z-dn.net/?f=%5Cfrac%7B%5Ccot%20x%7D%7B1%2B%5Ccsc%20x%7D%3D%5Cfrac%7B1-%5Csin%20x%7D%7B%5Ccos%20x%7D)
Divide both the numerator and denominator by sin x
![\frac{\cot x}{1+\csc x}=\frac{\frac{1}{\sin x}-\frac{\sin x}{\sin x}}{\frac{\cos x}{\sin x}}](https://tex.z-dn.net/?f=%5Cfrac%7B%5Ccot%20x%7D%7B1%2B%5Ccsc%20x%7D%3D%5Cfrac%7B%5Cfrac%7B1%7D%7B%5Csin%20x%7D-%5Cfrac%7B%5Csin%20x%7D%7B%5Csin%20x%7D%7D%7B%5Cfrac%7B%5Ccos%20x%7D%7B%5Csin%20x%7D%7D)
This finally gives:
![\frac{\cot x}{1+\csc x}=\frac{\csc x-1}{\cot x}](https://tex.z-dn.net/?f=%5Cfrac%7B%5Ccot%20x%7D%7B1%2B%5Ccsc%20x%7D%3D%5Cfrac%7B%5Ccsc%20x-1%7D%7B%5Ccot%20x%7D)
A 57.6 B. 100.8 c.78.2 I’m not sure
The mode is 4 because it appears the most
hope this helps!
Answer:
it should take the workers 18 days to finish
Step-by-step explanation:
Answer:
r= 2
Step-by-step explanation:
2^3 is basically 2x2x2
2x2=4 x 2= 8
you can subtract 10 with 8 to fin what r equals to.