SOLUTION: Please show me how to prove: (cscx-1)(1+cscx)=(cscx cosx)/(secx sinx)

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Question 1028156: Please show me how to prove:
(cscx-1)(1+cscx)=(cscx cosx)/(secx sinx)
Found 2 solutions by addingup, MathTherapy:
Answer by addingup(3677)   (Show Source): You can put this solution on YOUR website!
prove:
(cscx-1)(1+cscx)=(cscx cosx)/(secx sinx)
I'm going to rewrite it this way:
(csc(x)-1)(1+csc(x)) = (csc(x)cos(x))/(sec(x)sin(x))
Multiply both sides by sin(x) and by sec(x):
sec(x)sin(x)(csc(x)-1)(1+csc(x)) = ^?cos(x)csc(x)
Write cosecant as 1/sine and secant as 1/cosine:
1/(cos(x))sin(x)(1/(sin(x))-1)(1+1/(sin(x))) = ^?1/(sin(x))cos(x)
((1/(sin(x)))-1)(1+(1/(sin(x))))(1/(cos(x)))sin(x) = ((1/(sin(x))-1)(1+1/(sin(x)))sin(x))/(cos(x)):
(sin(x)(1/(sin(x))-1)(1+1/(sin(x))))/(cos(x)) = ^?cos(x)(1/(sin(x)))
Put 1/(sin(x))-1 over the common denominator sin(x): 1/(sin(x))-1 = (1-sin(x))/(sin(x)):
((1-sin(x))/(sin(x))sin(x)(1+1/(sin(x))))/(cos(x)) = ^?(cos(x))/(sin(x))
Put 1+1/(sin(x)) over the common denominator sin(x): 1+1/(sin(x)) = (1+sin(x))/(sin(x)):
((1+sin(x))/(sin(x))sin(x)(1-sin(x)))/(sin(x)cos(x)) = ^?(cos(x))/(sin(x))
Cancel sin(x) from the numerator and denominator. ((1-sin(x))(1+sin(x))sin(x))/(sin(x)sin(x)cos(x)) = (sin(x)(-(sin(x)-1)(1+sin(x))))/(sin(x)sin(x)cos(x)) = -((sin(x)-1)(1+sin(x)))/(sin(x)cos(x)):
-((sin(x)-1)(1+sin(x)))/(cos(x)sin(x)) = ^?(cos(x))/(sin(x))
Cross multiply:
-sin(x)(sin(x)-1)(1+sin(x)) = ^?cos(x)^2 sin(x)
Divide both sides by sin(x):
-((sin(x)-1)(1+sin(x))) = ^?cos(x)^2
-(sin(x)-1)(1+sin(x)) = 1-sin(x)^2:
1-sin(x)^2 = ^?cos(x)^2
cos(x)^2 = 1-sin(x)^2:
1-sin(x)^2 = ^?1-sin(x)^2
Look at the left hand side and the right hand side. They are identical, so the identity has been verified. The equality is true
Answer by MathTherapy(10553)   (Show Source): You can put this solution on YOUR website!
Please show me how to prove:
(cscx-1)(1+cscx)=(cscx cosx)/(secx sinx)
Usually, one side is chosen and then that side is proven to be equal to the other. With this, each side needs to be altered as follows:



LEFT side:



 ------- Rearranging binomial

 ------ FOILing binomials

 ------ Applying IDENTITY:  



RIGHT side:



 ------ Replacing 

__________ ----- 

 ------>  (PROVEN) 

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