This is for my trig class, and the directions say use the fundamental trigonometric identities to simplify the expression. Thanks for the help!

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

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