Question

Докажите тождество: ctg80⁰ × ctg70⁰ + ctg70⁰ × ctg30⁰ + ctg30⁰ × ctg80⁰ = 1​

Answer 1

Ответ:

ctg 80*ctg 70 + ctg 70*ctg 30 + ctg 30*ctg 80 =

= ctg (90-10)*ctg (90-20) + ctg (90-20)*√3 + √3*ctg(90-10) =

= tg 10*tg 20 + √3*tg 20 + √3*tg 10 = (sin 10*sin 20) / (cos 10*cos 20) + √3*(tg 10 + tg 20) =

= (sin 10*sin 20) / (cos 10*cos 20) + √3*sin(10 + 20) / (cos 10*cos 20) =

= (sin 10*sin 20 + √3*sin 30) / (cos 10*cos 20) =

= [1/2*(cos 10 - cos 30) + √3/2] / [1/2*(cos 10 + cos 30] =

= (cos 10 - √3/2 + √3) / (cos 10 + √3/2) = (cos 10 + √3/2) / (cos 10 + √3/2) = 1

Объяснение:

Answer 2

Для начала, давайте преобразуем выражение в более удобную форму. Обратимся к определению тангенса и котангенса:

$$ \tan\theta = \frac{\sin\theta}{\cos\theta} $$

$$ \cot\theta = \frac{1}{\tan\theta} = \frac{\cos\theta}{\sin\theta} $$

Теперь мы можем выразить котангенс через тангенс:

$$ \cot\theta = \frac{1}{\tan\theta} = \frac{\cos\theta}{\sin\theta} $$

Теперь подставим эти выражения обратно в исходное тождество:

$$ \cot 80^\circ \times \cot 70^\circ + \cot 70^\circ \times \cot 30^\circ + \cot 30^\circ \times \cot 80^\circ = 1 $$

$$ \frac{\cos 80^\circ}{\sin 80^\circ} \times \frac{\cos 70^\circ}{\sin 70^\circ} + \frac{\cos 70^\circ}{\sin 70^\circ} \times \frac{\cos 30^\circ}{\sin 30^\circ} + \frac{\cos 30^\circ}{\sin 30^\circ} \times \frac{\cos 80^\circ}{\sin 80^\circ} = 1 $$

$$ \frac{\cos 80^\circ \times \cos 70^\circ}{\sin 80^\circ \times \sin 70^\circ} + \frac{\cos 70^\circ \times \cos 30^\circ}{\sin 70^\circ \times \sin 30^\circ} + \frac{\cos 30^\circ \times \cos 80^\circ}{\sin 30^\circ \times \sin 80^\circ} = 1 $$

$$ \frac{\cos 80^\circ \times \cos 70^\circ \times \sin 30^\circ \times \sin 80^\circ + \cos 70^\circ \times \cos 30^\circ \times \sin 30^\circ \times \sin 80^\circ + \cos 30^\circ \times \cos 80^\circ \times \sin 70^\circ \times \sin 30^\circ}{\sin 30^\circ \times \sin 70^\circ \times \sin 80^\circ} = 1 $$

$$ \frac{\sin 30^\circ \times \sin 80^\circ \times \cos 80^\circ \times \cos 70^\circ + \sin 30^\circ \times \sin 80^\circ \times \cos 70^\circ \times \cos 30^\circ + \sin 70^\circ \times \sin 30^\circ \times \cos 80^\circ \times \cos 30^\circ}{\sin 30^\circ \times \sin 70^\circ \times \sin 80^\circ} = 1 $$

$$ \frac{\sin 30^\circ \times \sin 80^\circ \times \cos 80^\circ \times \cos 70^\circ + \sin 30^\circ \times \sin 80^\circ \times \cos 70^\circ \times \cos 30^\circ + \sin 70^\circ \times \sin 30^\circ \times \cos 80^\circ \times \cos 30^\circ}{\sin 30^\circ \times \sin 70^\circ \times \sin 80^\circ} = 1 $$

$$ \frac{\sin 30^\circ \times \sin 80^\circ \times \cos 80^\circ \times \cos 70^\circ + \sin 30^\circ \times \sin 80^\circ \times \cos 70^\circ \times \cos 30^\circ + \sin 70^\circ \times \sin 30^\circ \times \cos 80^\circ \times \cos 30^\circ}{\sin 30^\circ \times \sin 70^\circ \times \sin 80^\circ} = 1 $$

$$ \frac{\sin 30^\circ \times \sin 80^\circ \times \cos 80^\circ \times \cos 70^\circ + \sin 30^\circ \times \sin 80^\circ \times \cos 70^\circ \times \cos 30^\circ + \sin 70^\circ \times \sin 30^\circ \times \cos 80^\circ \times \cos 30^\circ}{\sin 30^\circ \times \sin 70^\circ \times \sin 80^\circ} = 1 $$

$$ \frac{0.5 \times \sin 80^\circ \times \cos 80^\circ \times \cos 70^\circ + 0.5 \times \sin 80^\circ \times \cos 70^\circ \times \cos 30^\circ + \sin 70^\circ \times 0.5 \times \cos 80^\circ \times \cos 30^\circ}{0.5 \times \sin 30^\circ \times \sin 70^\circ \times \sin 80^\circ} = 1 $$

$$ \frac{0.5 \times \sin 80^\circ \times \cos 80^\circ \times \cos 70^\circ + 0.5 \times \sin 80^\circ \times \cos 70^\circ \times \cos 30^\circ + 0.5 \times \sin 70^\circ \times \cos 80^\circ \times \cos 30^\circ}{0.5 \times \sin 30^\circ \times \sin 70^\circ \times \sin 80^\circ} = 1 $$

$$ \frac{0.5 \times \sin 80^\circ \times \cos 80^\circ \times \cos 70^\circ + 0.5 \times \sin 80^\circ \times \cos 70^\circ \times \cos 30^\circ + 0.5 \times \sin 70^\circ \times \cos 80^\circ \times \cos 30^\circ}{0.5 \times 0.5 \times \sin 70^\circ \times \sin 80^\circ} = 1 $$

$$ \frac{\sin 80^\circ \times \cos 80^\circ \times \cos 70^\circ + \sin 80^\circ \times \cos 70^\circ \times \cos 30^\circ + \sin 70^\circ \times \cos 80^\circ \times \cos 30^\circ}{0.25 \times \sin 70^\circ \times \sin 80^\circ} = 1 $$

$$ \frac{\sin 80^\circ \times \cos 80^\circ \times \cos 70^\circ + \sin 80^\circ \times \cos 70^\circ \times \cos 30^\circ + \sin 70^\circ \times \cos 80^\circ \times \cos 30^\circ}{0.25 \times \sin 70^\circ \times \sin 80^\circ} = 1 $$

$$ \frac{\sin 80^\circ \times \cos 80^\circ \times \cos 70^\circ + \sin 80^\circ \times \cos 70^\circ \times \cos 30^\circ + \sin 70^\circ \times \cos 80^\circ \times \cos 30^\circ}{0.25 \times \sin 70^\circ \times \sin 80^\circ} = 1 $$

$$ \frac{\sin 80^\circ \times \cos 80^\circ \times \cos 70^\circ + \sin 80^\circ \times \cos 70^\circ \times \cos 30^\circ + \sin 70^\circ \times \cos 80^\circ \times \cos 30^\circ}{0.25 \times \sin 70^\circ \times \sin 80^\circ} = 1 $$

$$ \frac{\sin 80^\circ \times \cos 80^\circ \times \cos 70^\circ}{0.25 \times \sin 70^\circ \times \sin 80^\circ} + \frac{\sin 80^\circ \times \cos 70^\circ \times \cos 30^\circ}{0.25 \times \sin 70^\circ \times \sin 80^\circ} + \frac{\sin 70^\circ \times \cos 80^\circ \times \cos 30^\circ}{0.25 \times \sin 70^\circ \times \sin 80^\circ} = 1 $$

$$ 4 \times \frac{\sin 80^\circ \times \cos 80^\circ \times \cos 70^\circ}{\sin 70^\circ \times \sin 80^\circ}