How do you prove `tanx + (cosx)/(1+cosx) = secx`?

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)`?