Question

Without using trigonometric tables, evaluate the follow
${\tan 1}^0{\tan 2}^0{\tan 3}^0........{tan89}^0$

Answer 1

$=tan1.tan2.tan3........tan88.tan89$

$=\frac{sin1}{cos1}.\frac{sin2}{cos2}....\frac{sin45}{cos45}.....\frac{sin88}{cos88}.\frac{sin89}{cos89}$

$=\frac{sin(90-89)}{cos1}.\frac{sin(90-88)}{cos2}...........\frac{sin(90-45)}{cos45}.........\frac{sin(90-1)}{cos89}$

$=\frac{cos89}{cos1}.\frac{cos88}{cos2}.....\frac{cos45}{cos45}...........\frac{cos1}{cos89}$

Finally all the values will get cancelled and the result will be $1$.