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Simplify 1/cscx+1 - 1/cscx-1

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1 Answer

Perhaps you wrote it improperly.  The answer is ZERO
 
(1/cscx - 1/cscx)  +  (1-1) = 0
 
Serenity - You need to be very careful with parenthesis.  I think you meant to pose the question as follows:
 
1/(cscx+1) - 1/(cscx-1)
 
= 1/(1/sinx+1) - 1/(1/sinx-1)
 
 
=sinx/(1+sinx) - sinx/(1-sinx)
=sinx( 1/(1+sinx) - 1/(1-sinx)  )
=sinx( (1-sinx) - (1+sinx) )/(1-sin^2x)
=sinx(-2sinx)/ (1-sin^2x)
=-2sin^2x/cos^2x
=-2tan^2x
 
Want an even easier method?
 
1/(cscx+1) - 1/(cscx-1)
 
=  ( cscx-1 - (cscx+1) )/ (csc^2x-1)
= -2/cot^2x      (because csc^2x-1 = cot^2x)
= -2tan^2x

Comments

 Two terms are opposite of each other, add up to 0.