How do you identify #1/(csctheta+1)-1/(csctheta-1)#?

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Answer 1 · 2017-04-19T10:10:17

#-2tan^2theta#

I'm going to assume this needs to be simplified:

#1/(csctheta+1)-1/(csctheta-1)#

#1/(1/sintheta+1)-1/(1/sintheta-1)#

#1/(1/sintheta+sintheta/sintheta)-1/(1/sintheta-sintheta/sintheta)#

#1/((1+sintheta)/sintheta)-1/((1-sintheta)/sintheta)#

#sintheta/(1+sintheta)-sintheta/(1-sintheta)#

#sintheta/(1+sintheta)((1-sintheta)/(1-sintheta))-sintheta/(1-sintheta)((1+sintheta)/(1+sintheta))#

#(sintheta(1-sintheta))/((1+sintheta)(1-sintheta))-(sintheta(1+sintheta))/((1-sintheta)(1+sintheta))#

#(sintheta-sin^2theta)/((1+sintheta)(1-sintheta))-(sintheta+sin^2theta)/((1-sintheta)(1+sintheta))#

#(sintheta-sintheta-sin^2theta-sin^2theta)/((1+sintheta)(1-sintheta))#

#(-2sin^2theta)/((1-sin^2theta))#

Recall that #sin^2theta+cos^2theta=1=>1-sin^2theta=cos^2theta#

#(-2sin^2theta)/cos^2theta=-2tan^2theta#