How do you find tan (x+y) if tan x=5/4 and sec y=2? Trigonometry Trigonometric Identities and Equations Proving Identities 1 Answer Ratnaker Mehta Jul 13, 2016 #tan(x+y)=(5+4sqrt3)/(4-5sqrt3),# or,#=(5-4sqrt3)/(4+5sqrt3)# Explanation: Given that #secy=2#, we use the Identity #: sec^2y=1+tan^2y# to get, #tany=+-sqrt(sec^2y-1) = +-sqrt(4-1)=+-sqrt3.# Now, #tan(x+y)=(tanx+tany)/(1-tanx*tany)=(5/4+-sqrt3)/(1-5/4*(+-sqrt3))# #=(5+-4sqrt3)/(4-(+-5sqrt3)# Thus, #tan(x+y)=(5+4sqrt3)/(4-5sqrt3),# or,#=(5-4sqrt3)/(4+5sqrt3)# 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 2972 views around the world You can reuse this answer Creative Commons License