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

Question

2323 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2015-11-14T15:26:39

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

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.
$1+color(green)(cottheta)$

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