cosx +sinxtanx=secx
tanx=sinx/cosx (quotient indentity)
cosx +sinx(sinx/cosx) =secx
cosx +sin^2x/cosx =secx
make a common denominator
cos^2x /cosx +sin^2x/cosx =secx
(cos^2x+sin^2x) /cosx =secx
cos^2x+sin^2x=1 (pythagorean identity)
1/cosx =sec x
1/cosx=secx (reciprocal identity)
secx=secx