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