Expert: Sherman D. - 2/8/2006

Question
How can I simplify the expression:   
(sec x + csc x) / (1 + tan x)

Answer
(secx + cscx)/(1 + tanx)

((1/cosx) + (1/sinx))/(1 + tanx)

Multiply the top by (cosx)(sinx)

((sinx + cosx)/((cosx)(sinx))/(1 + tan(x))

tan(x) as you know is (sinx)/(cosx)

((sinx + cosx)/((cosx)(sinx))/(1 + ((sinx)/(cosx)))

Multiply the bottom by cos(x)

((sinx + cosx)/((cosx)(sinx))/((cosx + sinx)/(cosx)))

Multiply the top half by the recipocal of the bottom half

((sinx + cosx)/((cosx)(sinx)) * (cosx/(cosx + sinx))

This gives you

((cosx)(sinx + cosx))/((cosx)(sinx)(cosx + sinx))

As you can see, you have (cosx + sinx) on top and bottom, so those cancel out, leaving you with

(cosx)/((cosx)(sinx))

as you can see, cosx cancels out, leaving you with

1/(sinx) or cscx

csc(x) is the answer.

there may be a shorter way, but that is the best way i know how to work it out.