How to do this trig identity?

Question

Tan o - sec o/ sin o

The Os are supposed to be thetas and the answer is -cot o.
Word form is tan theta minus sec theta over sin theta

Answer 1

Given tano - seco/sino

From trigonometric identities tano = sino/coso

Seco = 1/coso

Substitute in above equation

sino/coso - (1/coso)/sino

sino/coso - 1/(coso*sino)

Taking L . C. M , 

Sin^2o - 1/(coso*sino) ----------------(1)

Also we know that sin^2o + cos^2o = 1

Subtracting 1 - cos^2o from each side

sin^20 - 1 = - cos^2o

Subtitute sin^2o - 1 = - cos^2o in (1)

- cos ^2o/(coso*sino)

- coso/sino

- coto