how to prove the following: tan^2x+1+tanx secx= 1+sin x/ cos^2x

Question

i'm not sure how to proceed with this question. i missed a few days of the lesson and i am definitely floundering. all i have at my  disposal is a table of fundamental trigonemetric trig identities including reciprocal identities, tangent/ cotangent ratio identities, pythagorean identities and negative angle identites.

Answer 1

tanx*secx=sinx/cos^2x or sinx*sec^2x. tan^2x+1=sec^2x. So we have sec^2x+sec^2xsinx=sec^2x(1+sinx)=(1+sinx)/cos^2x. (To prove sec^2x=tan^2x+1, start with sin^2x+cos^2x=1, then divide through by cos^2x: tan^2x+1=sec^2x because secx is 1/cosx.)