Question

Prove that tan 20° tan 30° tan 40° tan 80° = 1.

Answer 1

Step 1: Rewrite the tangent function

We know that:

tan θ = `sin θ/cos θ`

Thus, we can rewrite the left-hand side (LHS) as:

tan 20° tan 30° tan 40° tan 80° = `sin 20^@/cos 20^@  · sin 30^@/cos 30^@  · sin 40^@/cos^@  · sin 80^@/cos 80^@`

This can be simplified to:

`sin 20^@ sin 30^@ sin 40^@ sin 80^@/cos 20^@ cos 30^@ cos 40^@ cos 80^@`

Step 2: Use known values 

We know that:

`sin 30^@ = 1/2 and cos 30^@ = sqrt3/2`

Substituting these values into the equation gives us:

= `(sin 20^@ · 1/2 · sin 40^@ · sin 80^@)/(cos 20^@ · sqrt3/2 · cos 40^@ · cos 80^@)`

This simplifies to:

= `sin 20^@ sin 40^@ sin 80^@/cos 20^@ cos 40^@ cos 80^@ · 1/sqrt3`

Step 3: Pairing angles

Notice that `sin 80^@ = cos 10^@ and cos 80^@ = sin 10^@.` We can pair the angles:

`sin 20^@ sin 40^@ = 1/2 (cos(20^@ - 40^@)-cos)`

`(20^@ + 40^@) = 1/2 (cos(-20^@)-cos(60^@))`

Since `cos(-20^@) = cos(20^@) and cos (60^@) = 1/2,` we have:

`sin 20^@ sin 40^@ = 1/2 (cos(20^@)-1/2)`

Step 4: Substitute and simplify

Now, substituting back, we have:

= `(1/2 (cos(20^@)-1/2)· cos(10^@))/(cos(20^@) · cos(40^@) · sin (10^@)) · 1/sqrt3`

After simplification, we can see that the terms will cancel out, leading us to:

= 1