#### Answer

$r' = tan^{2}\theta + cos\theta +1$
$r' = sec^{2}\theta + cos\theta$

#### Work Step by Step

$r = (1+sec\theta)sin\theta$
$r' = (sec\theta.tan\theta)sin\theta + (1+sec\theta)cos\theta$
$r' = tan^{2}\theta + cos\theta +1$
$r' = sec^{2}\theta + cos\theta$