How do I prove this is an identity? 1/csc(theta)/csc(theta)-cot(theta)=1+cos(theta)/sin(theta)

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Answer 1 · 2018-03-23T04:25:10

This is not an identity...

LHS #"" =>1/(csctheta/csctheta)-cottheta#

#=>1/(cancelcsctheta/cancelcsctheta)-cottheta#

#=>1/1-cottheta#

RHS#=>1+costheta/sintheta#

#=>1+cottheta#

#LHS !=RHS#

Answer 2 · 2018-03-23T08:20:05

See the explanation

#LHS = 1/(csctheta-cottheta#

Substitute 1 with #(csc^2theta-cot^2theta)#

#=> (csc^2theta-cot^2theta)/(csctheta-cottheta#

#=> ((csctheta-cottheta)(csctheta+cottheta))/(csctheta-cottheta#

#=> [(cancel(csctheta-cottheta))(csctheta+cottheta)]/cancel((csctheta-cottheta)#

#=> csctheta+cottheta#

#RHS = (1+costheta)/sintheta#

#=> 1/sintheta + costheta/sintheta#

#=> csctheta+cottheta#

Hence , #LHS=RHS#