Prove the identity

sin(w)·tan w + cos(w)        =      sec(w)

Simplify the more complicated side, the left
Use the identity tanq = sinq/cosq

Replace tan(w) by sin(w)/cos(w)

       sin(w)
sin(w)·—————— + cos(w)        =      sec(w)
       cos(w)


       sin²(w)  cos(w)   
      ——————— + ———————        =      sec(w)
       cos(w)     1

Get the LCD = cos(w)


sin²(w)   cos(w)·cos(w)   
——————— + —————————————        =      sec(w)
cos(w)       1·cos(w)

sin²(w)      cos²(w)   
——————— + —————————————        =      sec(w)
cos(w)       cos(w)

Write over the LCD 

sin²(w) +  cos²(w)   
——————————————————             =      sec(w)
     cos(w)  

Use identity sin²q + cos²q = 1

       1   
    ———————                    =      sec(w)
     cos(w)    


Use identity secq = 1/cosq

       sec(w)                  =      sec(w)

QED

Edwin
AnlytcPhil@aol.com