How do you verify #(tan^3(x) - 1) / (tan(x) - 1) = sec^2(x) + tan(x)#? Trigonometry Trigonometric Identities and Equations Proving Identities 1 Answer Abhinav T. Jun 29, 2016 LHS = #(tan^3(x)-1)/(tan(x)-1)# #"Since " a^3-b^3=(a-b)*(a^2+a*b+b^2)#, we may write the LHS #= (cancel((tan(x)-1))*(tan^2(x)+tan(x)+1))/cancel((tan(x)-1))# #=cancel(tan^2(x))+tan(x)+Sec^2(x)-cancel(tan^2(x))# #["Since " Sec^2(x)-tan^2(x) = 1]# #=Sec^2(x)+tan(x) = #RHS Proved. 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 13620 views around the world You can reuse this answer Creative Commons License