Help with simplifying two trigonometric identities (below)?

Question

1. tan(x)/csc(x)+cot(x)

2. sin(x)/1+cos(x)

Thank you in advance.

Answer 1

1.

tanx / cscx + cotx = tanx (cscx - cotx) / csc²x - cot²x

                               = tanx(cscx - cotx) / 1

Recall : Trigonometry identities csc²x - cot²x = 1

                               = tanx(cscx - cotx)

                               = 1/ cosx - tanx×cotx

                               = 1/ cosx - 1

                               = (1 - cosx)  / cosx

                               = secx - 1.  

Answer 2

2.

sinx / 1 + cosx

Recall : Trigonometric identities sinx = 2sin(x/2)cos(x/2) and 2cos²(x/2) -1 = cosx

sinx / 1 + cosx = 2sin(x/2)cos(x/2) / 2cos²(x / 2)

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

                          = tan(x / 2).