Explanation:
tan(π/8)
Using half angle formula:
tan(θ/2) = (1 − cos θ) / sin θ
tan(π/8) = (1 − cos(π/4)) / sin(π/4)
= (1 − √2/2) / (√2/2)
= (2 − √2) / √2
= (2√2 − 2) / 2
= √2 − 1
If you want to prove it using the formula you derived, use A = π/8.
2A + B = π/4
2(π/8) + B = π/4
B = 0
Therefore:
tan B = (1 − 2 tan A − tan² A) / (1 + 2 tan A − tan² A)
tan 0 = (1 − 2 tan(π/8) − tan²(π/8)) / (1 + 2 tan(π/8) − tan²(π/8))
0 = 1 − 2 tan(π/8) − tan²(π/8)
tan²(π/8) + 2 tan(π/8) − 1 = 0
tan²(π/8) + 2 tan(π/8) + 1 = 2
(tan(π/8) + 1)² = 2
tan(π/8) + 1 = ±√2
tan(π/8) = ±√2 − 1
Since π/8 is in the first quadrant, we know tan(π/8) > 0.
tan(π/8) = √2 − 1
sin 20 sin 40 sin 60 sin 80
sin 60 = √3/2, so:
√3/2 (sin 20 sin 40 sin 80)
We can use product to sum formula to simplify. For the first attempt, let's use the product of the first two terms.
√3/2 (sin 20 sin 40) sin 80
√3/4 (cos(40−20) − cos(40+20)) sin 80
√3/4 (cos 20 − cos 60) sin 80
√3/4 (cos 20 − 1/2) sin 80
√3/8 (2 cos 20 − 1) sin 80
Distribute:
√3/8 (2 cos 20 sin 80 − sin 80)
Repeat product to sum formula:
√3/8 (sin(80+20) + sin(80−20) − sin 80)
√3/8 (sin 100 + sin 60 − sin 80)
√3/8 (sin 100 + √3/2 − sin 80)
√3/8 (sin(180−80) + √3/2 − sin 80)
Use angle difference formula:
√3/8 (sin 180 cos 80 − sin 80 cos 180 + √3/2 − sin 80)
√3/8 (sin 80 + √3/2 − sin 80)
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That's one way. Let's do it another way. This time, we'll use the product of the last two terms.
√3/2 sin 20 (sin 40 sin 80)
Product to sum:
√3/4 sin 20 (cos(80−40) − cos(80+40))
√3/4 sin 20 (cos 40 − cos 120)
√3/4 sin 20 (cos 40 + 1/2)
√3/8 sin 20 (2 cos 40 + 1)
Distribute:
√3/8 (2 cos 40 sin 20 + sin 20)
Repeat product to sum:
√3/8 (sin(40+20) − sin(40−20) + sin 20)
√3/8 (sin 60 − sin 20 + sin 20)
√3/8 sin 60
√3/8 (√3/2)
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