Question

tan5 tan25 tan30 tan65 tan85=?

Answer 1

tan(90-85)*tan(90-65)*tan30*tan65*tan85

cot 85*cot65*tan30*tan65* tan85

(1/tan85)*(1/tan65)*tan30*tan65* tan85

tan30=1/√3

Answer 2

tan(90 - x)

= sin(90 - x) ÷ cos(90 - x)

= (sin90cosx - cos90sinx) ÷ (cos90cosx + sin90sinx)

= (1cosx - 0sinx) ÷ (0cosx + 1sinx)

= (cosx) ÷ (sinx)

= cosx/sinx

= cotx

= 1/tanx

so,

tan(x) × tan(90 - x) = 1 or we can say that:

tan(A) × tan(B) = 1

where A + B = 90 degrees

and A and B are positive and does not equal to 90 nor 0

**********

5 + 85 = 90

25 + 65 = 90

tan(5) tan(25) tan(30) tan(65) tan(85)

= tan(5)tan(85) × tan(25)tan(65) × tan(30)

= 1 × 1 × tan(30)

= tan(30)

= 1/√3

where x does not equal to 0 or 90