please help me with more identities?

Question

steps and work prove identitites

1. 1/(sinx+1) + 1/ (cscx+1) =1

2. cos x - (cos x/(1-tanx) =( sinxcosx/ sinx - cos x)

3. tan(pi/2 -x)tan x = 1

4. cos bracket symbol(pi/2)-x bracket symbol over sin bracket symbol(pi/2) -x bracket symbol = tan x

csc(-x) / sec(-x) = -cot x

Answer 1

1. 1/(sinx+1) + 1/ (cscx+1) =1
1/(sinx +1) + 1/(1/sinx +1) = 1
1/(sinx +1) +1/(1+sinx)/sinx)=1
1/(sinx +1) + sinx/(sinx +1) = 1
(1=sinx)/(sinx+1) = 1
1 = 1
2. cos x - (cos x/(1-tanx) =( sinxcosx/ sinx - cos x)
cosx - cosx/(1- sinx/cosx) = ( sinxcosx/ sinx - cos x)
(cosx -sinx -cosx)/(cosx -sinx/cosx) =( sinxcosx/ sinx - cos x)
(-sinxcosx/(cosx -sinx) = ( sinxcosx/ sinx - cos x)
sinx cosx/(sinx -cosx = (sinxcosx/ sinx - cos x)

3. tan(pi/2 -x)tan x = 1
= sin(pi/2-x)/cos(pi/2-x)(sin x/cosx =1
= (cosx/sinx) sinx/cosx) = 1
1 =1
4. cos bracket symbol(pi/2)-x bracket symbol over sin bracket symbol(pi/2) -x bracket symbol = tan x
Same as prob 3.

csc(-x) / sec(-x) = -cot x
1/sin(-x)/1/cos(-x) = cot(x)
cos(-x)/sin(-x) = cot x
-cosx/-sinx= cotx
cotx=cotx

Answer 2

Here is the trick:

CONVERT everything to sine and cosine.

Then simplify.

Guido