Question

tan 20° tan 40° tan 60° tan 80° is equal to ______.

विकल्प

• 1

• 2

• 3

• `sqrt(3)/2`

Answer 1

tan 20° tan 40° tan 60° tan 80° is equal to 3.

Explanation:

tan 20° tan 40° tan 60° tan 80°

= `(sin 20^circ  sin 40^circ  sin 80^circ  tan 60^circ)/(cos 20^circ cos 40^circ cos 80^circ)`

Here N^r = (sin 20° sin 40° sin 80°)

= `(sin 20^circ)/2 (2  sin 40^circ sin 80^circ)`

= `(sin 20^circ)/2 (cos 40^circ -  cos 120^circ)`

= `1/2 sin 20^circ (1 - 2 sin^2 20^circ + 1/2)`

= `1/2 sin 20^circ (3/2 - 2 sin^2 20^circ)`

= `(sin 60^circ)/4`

= `sqrt(3)/8`

Now, we take D^r = cos 20° cos 40° cos 80°

= `(sin 2^3  20^circ)/(2^3  sin 20^circ)`

= `(sin 160^circ)/(8  sin 20^circ)`

= `(sin 20^circ)/(8  sin 20^circ)`

= `1/8`

Hence, tan 20° tan 40° tan 80° = `(sqrt(3)/8)/(1/8)`

Therefore, tan 20° tan 40° tan 60° tan 80° = `sqrt(3) . sqrt(3)` = 3