How do you prove #((1+cosx) / sinx) + (sinx / (1 + cosx)) = 2 csc x#? Trigonometry Trigonometric Identities and Equations Proving Identities 1 Answer Becca M. · Tiago Hands Apr 15, 2015 #LHS# #=(1+cosx)/sinx + sinx/(1+cosx)# #=((1+cosx)^2+sin^2x)/(sinx (1+cosx))# #=(1+2cosx + cos^2x + sin^2x)/(sinx(1+cosx))# #=(2+2cosx)/(sinx(1 + cosx))# #=(2(1+cosx))/(sinx(1+cosx))# #=2/sinx# #=2cscx# #=RHS# This is because: #a/b+c/d = (ad+bc)/(bd)# And also because: #cos^2x+sin^2x=1# Source here. Answer link Related questions What does it mean to prove a trigonometric identity? How do you prove #\csc \theta \times \tan \theta = \sec \theta#? How do you prove #(1-\cos^2 x)(1+\cot^2 x) = 1#? How do you show that #2 \sin x \cos x = \sin 2x#? is true for #(5pi)/6#? How do you prove that #sec xcot x = csc x#? How do you prove that #cos 2x(1 + tan 2x) = 1#? How do you prove that #(2sinx)/[secx(cos4x-sin4x)]=tan2x#? How do you verify the identity: #-cotx =(sin3x+sinx)/(cos3x-cosx)#? How do you prove that #(tanx+cosx)/(1+sinx)=secx#? How do you prove the identity #(sinx - cosx)/(sinx + cosx) = (2sin^2x-1)/(1+2sinxcosx)#? See all questions in Proving Identities Impact of this question 27287 views around the world You can reuse this answer Creative Commons License