Let #/_QCD=alpha#

#/_PCB=beta#

So #/_PCQ=90^@-(alpha+beta)#

Now for #Delta QCD,/_QDC=90^@ and DC = 1unit#

So #DQ=DCtanalpha=tanalpha#

And for #Delta PCB,/_PBC=90^@ and BC = 1unit#

So #PB=BCtanbeta=tanbeta#

Hence for #DeltaAPQ#

#AQ=1-tanalpha#

#AP=1-tanbeta#

and

#PQ=sqrt((1-tanalpha)^2+(1-tanbeta)^2)#

Given

#AQ+AP+PQ=2#

#=>1-tanalpha+1-tanbeta+sqrt((1-tanalpha)^2+(1-tanbeta)^2)=2#

#=>sqrt((1-tanalpha)^2+(1-tanbeta)^2)=tanalpha +tanbeta#

#=>1-2tanalpha+tan^2alpha+1-2tanbeta+tan^2beta=tan^2alpha+tan^2beta+2tanalphatanbeta#

#=>1-tanalpha-tanbeta=tanalphatanbeta#

#=>tanalpha+tanbeta=1-tanalphatanbeta#

#=>(tanalpha+tanbeta)/(1-tanalphatanbeta)=1=tan45^@#

#=>tan(alpha+beta)=tan45^@#

#=>alpha +beta=45^@#

Hence #/_PCQ=90^@-(alpha+beta)=90^@-45^@=45^@#

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