Please help me with my trig work?

Question

tan x-cot x/tan x+cot x
sin x(cot x-csc x)
(cot x+csc x)(1-cosx)

please simplify

Answer 1

sinx(cotx - cscx)

Recall : Trigonometry formulas cotx =cosx / sinx - 1 / sinx

sinx(cosx / sinx - 1 / sinx)

Recall : ditributive property a(b - c) = ab - ac

sinx(cosx / sinx) - sinx / sinx

cosx - 1.

 

 

Answer 2

(cotx + cscx)(1 - cosx)

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

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

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

Recall :(a - b)( a + b) = a² - b²

1 -cos²x / sinx

Recall : trigonometry formula

sin²x / sinx

sinx.

Answer 3

tanx - cotx / tanx + cotx

tanx - cot²x + cotx

Recall :Trigonometry formula tanx = sinx/cosx and cotx = cosx/sinx

(sinx/cosx) - (cos²x/sin²x)  + (cosx/sinx)

Using L.C.M

(sin³x - cos³x + sinx cos²x) / sin²xcosx

Recall : trigonometry formula sin²x + cos²x = 1 then cos²x = 1 -sin²x

[sin³x - cos³x + sinx(1 - sin²x)] / sin²xcosx

[sin³x - cos³x + sinx - sin³x] / sin²xcosx

Recall : Trigonometry formula

(sinx - cos³x) / cosx(1 - cos²x)

(sinx - cos³x) / (cosx - cos³x).