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

Question

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

Answer 1

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