Rewrite the expression in terms of sine and cosine, and simplify as much as possible. (sec w (1 + csc^2 w)) / (csc^2 w)

(sin^2 w + 1) / cos w.

Step-by-step explanation:

Note: sec w = 1 / cos w and csc w = 1 / sin w.

So we have:

(sec w (1 + csc^2 w)) / (csc^2 w)

= 1/cos w (1 + 1 / sin^2 w) / (1 / sin^2 w)

= (1 / cos w + 1 / sin^2 w cos w) * sin^2 w

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

= sin^2 w / cos w + 1 / cos w

= (sin^2 w + 1) / cos w.