Question

How do you simplify `csc theta [sin theta + (1/sec theta)]`?

Answer 1

Rewrite everything in terms of `sintheta` and `costheta`.

Explanation:

Know the following trigonometric identities:
`sintheta=1/csctheta,color(red)(csctheta=1/sintheta)`

`color(blue)(costheta=1/sectheta),sectheta=1/costheta`

`tantheta=sintheta/costheta,color(green)(cottheta=costheta/sintheta)`

Original:
`color(red)(csctheta)[sintheta+(color(blue)(1/sectheta))]`

Rewrite everything as `sintheta` and `costheta`.
Our new statement is:
`color(red)(1/sintheta)[sintheta+color(blue)(costheta)]`

Now, distribute.
`sintheta/sintheta+color(green)(costheta/sintheta)`

Simplify using identities.
`1+color(green)(cottheta)`