Solving by the Rule of Cross Multiplication, we get
pcschz+qsechz+r=0
p′cschz+q′sechz+r′=0
sechz 1 cschz
q r p q
q′ r′ p′ q′
cschzqr′−q′r=sechzrp′−r′p=1pq′−qp′
⇒cschzqr′−q′r=1pq′−qp′ and sechzrp′−r′p=1pq′−qp′
⇒cschz=qr′−q′rpq′−qp′ and sechz=rp′−r′ppq′−qp′
sinhz=pq′−qp′qr′−q′r
and coshz=pq′−qp′rp′−r′p
Squaring and subtracting we get
cos2hz−sin2hz=(pq′−qp′rp′−r′p)2−(pq′−qp′qr′−q′r)2
⇒(pq′−qp′rp′−r′p)2−(pq′−qp′qr′−q′r)2=1