Question

How would you prove `tanx/cscx = secx - 1/secx`?

Answer 1

Replace all simple functions
`tan x/csc x = tanx sin x = sin^2x/cos x`
`1/cosx - 1/secx = 1/cosx - cos x = sin^2x/cosx`

Answer 2

Just to clarify the answer of the previous contributor.

Explanation:

On the left side, we must apply the identities `tanx = sinx/cosx` and `cscx= 1/sinx`

`(sinx/cosx)/(1/sinx) = 1/cosx - 1/(1/cosx)`

`(sinx xx sinx)/cosx = 1/cosx - cosx`

`sin^2x/cosx = (1 - cos^2x)/cosx`

Applying the Pythagorean identity `1 - cos^2x = sin^2x`

`sin^2x/cosx = sin^2x/cosx`

Identity proved!!

Hopefully this helps!