How to prove the trigonometric identity?

Question

#(1+sintheta-costheta)/(1+sintheta+costheta)+(1+sintheta+costheta)/(1+sintheta-costheta)=2cosectheta#

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Answer 1 · 2018-06-04T03:37:56

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

#=((1+sintheta-costheta)^2+(1+sintheta+costheta)^2)/((1+sintheta+costheta)(1+sintheta-costheta))#

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

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

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

#=(4(1+sintheta))/(2sintheta(1+sintheta))#

#=2/sintheta#

#=2cosectheta=RHS#