How do you prove (cos[theta]+tan[theta])/(sin[theta])= sec[theta]+cot[theta]?

2 Answers
Noah G
May 24, 2016

(costheta + sin theta/costheta)/sintheta = 1/costheta + costheta/sintheta

((cos^2theta + sin theta)/costheta)/sintheta = (sin theta + cos^2theta)/(costhetasintheta)

(cos^2theta + sin theta)/(sinthetacostheta) = (sin theta + cos^2theta)/(costhetasintheta)

Identity proved!!

Hopefully this helps!

Truong-Son N.
May 24, 2016

Here's an alternative way to do it.

Recall that sectheta = 1/costheta, and cottheta = 1/(tantheta) = costheta/sintheta.

So, multiplying by sintheta gives:

(costheta + tantheta)/cancel((sintheta))*cancel(sintheta) = sinthetasectheta + sinthetacottheta

costheta + tantheta = sintheta*1/costheta + cancel(sintheta)*costheta/cancel(sintheta)

color(blue)(costheta+tantheta = tantheta + costheta)

Related questions
See all questions in Proving Identities
Impact of this question
4919 views around the world
You can reuse this answer
Creative Commons License