By converting into parametric equations,
#{(x(theta)=r(theta)cos theta=(1+sin theta)cos theta),(y(theta)=r(theta)sin theta=(1+sin theta)sin theta):}#
By finding the derivatives using Product Rule,
#x'(theta)=cos theta cdot cos theta + (1+sin theta)cdot(-sin theta)#
#=(cos^2theta-sin^2theta)-sin theta#
#=cos2theta-sin theta#
#Rightarrow x'(pi/4)=cos(pi/2)-sin(pi/4)=-1/sqrt{2}#
#y'(theta)=cos theta cdot sin theta+(1+sin theta)cdot cos theta#
#=2sin theta cos theta+cos theta#
#=sin2theta+cos theta#
#Rightarrow y'(pi/4)=sin(pi/2)+cos(pi/4)=1+1/sqrt{2}#
The slope #m# we are looking for is:
#m={dy}/{dx}|_{theta=pi/4}={y'(pi/4)}/{x'(pi/4)}={1+1/sqrt{2}}/{-1/sqrt{2}}
=-(sqrt{2}+1)#
I hope that this was helpful.
