Question 173390: Please help me verify this equation:
cscx - cscx
------ ------ =2sec^2x
1+cscx 1-cscx
yes, the cscx's are over 1+cscx and 1-cscx. Those are fractions.
So far i have gotten stuck on this porblem. i have expanded csc x into 1/sinx in both parts of the left side of the equation which gets me:
1/sinx - 1/sinx
-------- --------- = 2sec^2x
1+1/sinx 1-1/sinx
After that i tried to achieve a common denominator on the left side which is where i get lost. I'm not sure what step to do after that or how to solve it.
Answer by gonzo(654)   (Show Source):
You can put this solution on YOUR website! good one.
maybe i got the answer.
you want to prove this equation is true.
here goes:
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let c = csc(x)
let s = sec(x)
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your equation becomes:
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if you multiply both sides of the equation by (1-c)*(1+c), you get:
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simplifying, this becomes:
which becomes:
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this is where substitutions come in.
you know that:
you also know that:
sin(x) = 1/csc(x)
and that:
cos(x) = 1/sec(x)
substituting for sin(x) and cos(x), the equation becomes:
if you multiply both sides of this equation by  , the equation becomes:
subtract  from both sides of this equation to get:
simplify this on the right hand side to become:
divide both sides of this equation by  to get:
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this is the same as:
and, since we originally let c = csc(x) and we let s = sec(x), this equation now becomes:
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we can now substitute for c^2 in the original equation we derived above, which was:
simplifying this, it becomes:
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substituting for  , that equation becomes:
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if we multiply both sides of this equation by  , we get:
this becomes:
which becomes:
if we multiply both sides of this equation by (-1), it becomes:
which proves that the original equation of:
is true.
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since we originally substituted c for csc(x), and s for sec(x), the original equation becomes:
and the proven identify becomes:
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