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)
Answer:
TU = 29
Step-by-step explanation:
TU + UV = TV , substitute values
9x + 2 + 5 = 14x - 8 , that is
9x + 7 = 14x - 8 ( subtract 14x from both sides )
- 5x + 7 = - 8 ( subtract 7 from both sides )
- 5x = - 15 ( divide both sides by - 5 )
x = 3
Then
TU = 9x + 2 = 9(3) + 2 = 27 + 2 = 29
Answer:
Raul
Step-by-step explanation:
11/2 from 17.6 = 17.6 divided 2 by 11= 94.6
2/1 from 14.2 = 14.2 divided 1 by 2= 28.4
Raul did 94.6 and Ulee 28.4, Raul did more bike.
In order to determine volume you should multiple w*d*h. So in this case 5*6*3 or 90 cubic meters.