tan 10° × tan 15° × tan 75° × tan 80°

= tan 10 × tan 15 × tan(90-75) × tan (90-80)
= tan10 × tan15 × cot 15 × cot10
As we know,

tan θ × cot θ = 1
So,
tan 10 × cot 10 × tan 15 × cot 15
= 1