How do you verify the identity $sintheta/(1-costheta)+(1-costheta)/sintheta=2csctheta$ ?

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Answer 1 · 2016-08-15T16:34:37

$LHS=sintheta/(1-costheta)+(1-costheta)/sintheta$

$=(sintheta*sintheta)/((1-costheta)sintheta)+(1-costheta)/sintheta$

$=sin^2theta/((1-costheta)sintheta)+(1-costheta)/sintheta$

$=(1-cos^2theta)/((1-costheta)sintheta)+(1-costheta)/sintheta$

$=((1-costheta)(1+costheta))/((1-costheta)sintheta)+(1-costheta)/sintheta$

$=(1+costheta)/sintheta+(1-costheta)/sintheta$

$=(1+costheta+1-costheta)/sintheta$

$=2/sintheta=2csctheta=RHS$

Verified

How do you verify the identity sintheta/(1-costheta)+(1-costheta)/sintheta=2cscthetasintheta/(1-costheta)+(1-costheta)/sintheta=2csctheta?