Whakaoti i te (a^2+1)(c^2+1)=(ac-1)^2+(a+c)^2

$$( a ^ { 2 } + 1 ) ( c ^ { 2 } + 1 ) = ( a c - 1 ) ^ { 2 } + ( a + c ) ^ { 2 }$$

Whakaoti mō a

$$a\in \mathrm{R}$$

Ngā Upane Otinga

Whakamahia te āhuatanga tohatoha hei whakarea te a^{2}+1 ki te c^{2}+1.

$$a^{2}c^{2}+a^{2}+c^{2}+1=\left(ac-1\right)^{2}+\left(a+c\right)^{2}$$

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(ac-1\right)^{2}.

$$a^{2}c^{2}+a^{2}+c^{2}+1=a^{2}c^{2}-2ac+1+\left(a+c\right)^{2}$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(a+c\right)^{2}.

$$a^{2}c^{2}+a^{2}+c^{2}+1=a^{2}c^{2}-2ac+1+a^{2}+2ac+c^{2}$$

Pahekotia te -2ac me 2ac, ka 0.

$$a^{2}c^{2}+a^{2}+c^{2}+1=a^{2}c^{2}+1+a^{2}+c^{2}$$

Tangohia te a^{2}c^{2} mai i ngā taha e rua.

$$a^{2}c^{2}+a^{2}+c^{2}+1-a^{2}c^{2}=1+a^{2}+c^{2}$$

Pahekotia te a^{2}c^{2} me -a^{2}c^{2}, ka 0.

$$a^{2}+c^{2}+1=1+a^{2}+c^{2}$$

Tangohia te a^{2} mai i ngā taha e rua.

$$a^{2}+c^{2}+1-a^{2}=1+c^{2}$$

Pahekotia te a^{2} me -a^{2}, ka 0.

$$c^{2}+1=1+c^{2}$$

Whakaraupapatia anō ngā kīanga tau.

$$\text{true}$$

He pono tēnei mō tētahi a ahakoa.

$$a\in \mathrm{R}$$

Whakaoti mō c

$$c\in \mathrm{R}$$

Ngā Upane Otinga

Whakamahia te āhuatanga tohatoha hei whakarea te a^{2}+1 ki te c^{2}+1.

$$a^{2}c^{2}+a^{2}+c^{2}+1=\left(ac-1\right)^{2}+\left(a+c\right)^{2}$$

Whakamahia te ture huarua \left(p-q\right)^{2}=p^{2}-2pq+q^{2} hei whakaroha \left(ac-1\right)^{2}.

$$a^{2}c^{2}+a^{2}+c^{2}+1=a^{2}c^{2}-2ac+1+\left(a+c\right)^{2}$$

Whakamahia te ture huarua \left(p+q\right)^{2}=p^{2}+2pq+q^{2} hei whakaroha \left(a+c\right)^{2}.

$$a^{2}c^{2}+a^{2}+c^{2}+1=a^{2}c^{2}-2ac+1+a^{2}+2ac+c^{2}$$

Pahekotia te -2ac me 2ac, ka 0.

$$a^{2}c^{2}+a^{2}+c^{2}+1=a^{2}c^{2}+1+a^{2}+c^{2}$$

Tangohia te a^{2}c^{2} mai i ngā taha e rua.

$$a^{2}c^{2}+a^{2}+c^{2}+1-a^{2}c^{2}=1+a^{2}+c^{2}$$

Pahekotia te a^{2}c^{2} me -a^{2}c^{2}, ka 0.

$$a^{2}+c^{2}+1=1+a^{2}+c^{2}$$

Tangohia te c^{2} mai i ngā taha e rua.

$$a^{2}+c^{2}+1-c^{2}=1+a^{2}$$

Pahekotia te c^{2} me -c^{2}, ka 0.

$$a^{2}+1=1+a^{2}$$

Whakaraupapatia anō ngā kīanga tau.

$$\text{true}$$

He pono tēnei mō tētahi c ahakoa.

$$c\in \mathrm{R}$$