Tanx/cscx + sinx/tanx

Tan (x) = Sin (x) / Cos (x)

Csc (x) = 1/sin (x)

therefore: [sin (x) / cos (x) ] / [1/sin (x) ] + sin (x) / [sin (x) / cos (x) ]

but division is just multiplication by reciprocal

= sin (x) / cos (x). sin (x) + sin (x). cos (x) / sin (x)

= sin^2 (x) / cos (x) + cos (x)

=[sin^2 (x) + cos^2 (x) ] / cos (x) - - - since sin^2 (x) + cos^2 (x) = 1

= 1/cos (x)

= sec (x)