How to verify that these are identities

I'm sorry i'm asking a lot of questions, but a lot of these websites are confusing me.. thanks if you help I really need to understand this stuff.

1/sinx - sinx = cos^2x/sinx ?

tanx + 1 / tanx = 1 / sinx cosx?

1.) Lets get a common denominator

1/sinx - sinx =

1/sinx -sin^2x/sinx =

(1-sin^2x)/sinx =

cos^2x/sinx.


2.) Write in terms of cosine/sine and get a common denimonator

tanx + 1/tanx

sinx/cosx + 1/sinx/cosx

sinx/cosx + cosx/sinx

sin^2x/sinxcosx + cos^2x/sinxcosx

[sin^2x+ cos^2x]/sinxcosx

1/sinxcosx

You use the standard identities to show that one side equals the other
lhs
=1/sin(x) - sin(x)
= (1 - sin^2(x))/sin(x)
..... standard identity required is sin^2 + cos^2 = 1 so cos^2 = 1 - sin^2
=cos^2(x)/sin(x)
=rhs

lhs
= tan(x) + 1/tan(x)
= sin(x)/cos(x) + cos(x)/sin(x)
= (sin^2(x) + cos^2(x))/(sin(x)cos(x))
= 1/(sin(x)cos(x))
=rhs