How do you prove $(sectheta - tantheta) (csctheta +1)=cottheta$ ?

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Answer 1 · 2016-07-05T01:20:28

The following identities will be necessary for this problem:

• $sectheta = 1/costheta$

• $csctheta = 1/sintheta$

• $tantheta = sin theta/costheta$

• $cottheta = costheta/sintheta$

• $1 -sin^2theta = cos^2theta$

Now, we have what we need to prove:

$(1/costheta - sintheta/costheta)(1/sintheta + 1) = costheta/sintheta$

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

$(1 - sin^2theta)/(costhetasintheta) = costheta/sintheta$

$cos^2theta/(costhetasintheta) = costheta/sintheta$

$costheta/sintheta = costheta/sintheta$

Identity proved!!

Practice exercises:

Prove the following identities:

a) $sintheta + cos^2theta/(1 + sin theta) = 1$

b) $sintheta(csctheta - sin theta) = cos^2theta$

Hopefully this helps and good luck!

How do you prove (sectheta - tantheta) (csctheta +1)=cottheta(sectheta - tantheta) (csctheta +1)=cottheta?