SOLUTION: Help is super appreciated :) Establish the identity: {{{(csc(x)cos(x))/(csc(x)-sin(x))=sec(x)}}} (cscx*cosx)/(cscx-sinx)=secx

Algebra ->  Equations -> SOLUTION: Help is super appreciated :) Establish the identity: {{{(csc(x)cos(x))/(csc(x)-sin(x))=sec(x)}}} (cscx*cosx)/(cscx-sinx)=secx      Log On


   

Question 1154792: Help is super appreciated :)
Establish the identity:
%28csc%28x%29cos%28x%29%29%2F%28csc%28x%29-sin%28x%29%29=sec%28x%29
(cscx*cosx)/(cscx-sinx)=secx
Found 3 solutions by rothauserc, MathLover1, Edwin McCravy:
Answer by rothauserc(4718) About Me  (Show Source):
You can put this solution on YOUR website!
(csc(x) * cos(x))/(csc(x) -sin(x)) = sec(x)
:
Note csc(x) = 1/sin(x), sec(x) = 1/cos(x)
:
Work on the left side of the = sign
:
(cos(x)/sin(x)) / (((1/sin(x)) - sin(x)) =
:
(cos(x)/sin(x)) / ((1 - sin^2(x))/sin(x)) =
:
(cos(x)/sin(x)) * (sin(x)/(1 - sin^2(x)) =
:
cos(x)/cos^2(x) =
:
Note cos^2(x) = 1 - sin^2(x)
:
1/cos(x) = sec(x)
:

Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

%28csc%28x%29%2Acos%28x%29%29%2F%28csc%28x%29-sin%28x%29%29=sec%28x%29
manipulate left side
%28csc%28x%29%2Acos%28x%29%29%2F%28csc%28x%29-sin%28x%29%29 .......use identity: cos%28x%29=1%2Fsec%28x%29 and sin%28x%29=1%2Fcsc%28x%29

=%28csc%28x%29%2A%281%2Fsec%28x%29+%29%29%2F%28csc%28x%29-1%2Fcsc%28x%29%29+

=%28csc%28x%29%2Fsec%28x%29+%29%2F%28%28csc%5E2%28x%29-1%29%2Fcsc%28x%29%29 .........%28csc%5E2%28x%29-1%29=cot%5E2%28x%29

=%28csc%28x%29%2Fsec%28x%29+%29%2F%28cot%5E2%28x%29%2Fcsc%28x%29%29+

= csc%5E2%28x%29%2F%28sec%28x%29%2Acot%5E2%28x%29%29 .........

=

= 1%2F%281%2Fsec%28x%29%29+

= sec%28x%29


Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!

%28csc%28x%29cos%28x%29%29%2F%28csc%28x%29-sin%28x%29%29=sec%28x%29 

We'll work with the left side.

%28expr%281%2Fsin%28x%29%29cos%28x%29%29%2F%28expr%281%2Fsin%28x%29%29-sin%28x%29%29 

Multiply top and bottom by sin(x)

 

Cancel the sin(x)'s on top, distribute on the bottom:

 

Cancel the sin(x)'s in the first term on the bottom:



cos%5E%22%22%28x%29%2F%281-sin%5E2%28x%29%29

cos%5E%22%22%28x%29%2Fcos%5E2%28x%29

Cancel the cos(x) into the cos²(x)

cross%28cos%5E%22%22%28x%29%29%2Fcos%5Ecross%282%29%28x%29

1%2Fcos%28x%29

sec%28x%29

Edwin