Question

Найдите выражение (cos 75 - sin 75) ^2​

Answer 1

$\displaystyle\bf\\\Big(Cos75^\circ-Sin75^\circ\Big)^{2}=Cos^{2} 75^\circ-2Sin75^\circ Cos75^\circ+Sin^{2} 75^\circ=\\\\=\underbrace{\Big(Cos^{2} 75^\circ+Sin^{2} 75^\circ\Big)}_{1}-\underbrace{2Sin75^\circ Cos75^\circ}_{Sin150^\circ}=1-Sin150^\circ=\\\\=1-Sin(180^\circ-30^\circ)=1-Sin30^\circ=1-0,5=0,5$

Answer 2

Раскроем скобки:

$(cos75-sin75)^{2} =cos^{2} 75 - 2cos75sin75 + sin^{2}75$

Квадрат косинуса представим в виде разности единицы и квадрата синуса по ОТТ:

$cos^{2} 75 = 1 -sin^{2}75$

Запишем полученное выражение:

$1 - sin^{2} 75 - 2sin75cos75 + sin^{2}75 = 1 -2sin75cos75$

Заметим, что тут спрятана формула синуса двойного угла:

sin2x = 2sinxcosx

Выражение принимает вид:

1 - sin2*75 = 1 - sin150

sin150 = 0,5 - табличная величина

1 - 0,5 = 0,5

Ответ: 0,5

Answer 3

I can help you with math problems. Here is the detailed solution for your question:

The expression $(\cos 75 - \sin 75)^2$ can be simplified by using the following trigonometric identities:

- $\cos(A-B) = \cos A \cos B + \sin A \sin B$ - $\sin(A-B) = \sin A \cos B - \cos A \sin B$

Using these identities, we can write:

$$\cos 75 = \cos(90-15) = \sin 15$$ $$\sin 75 = \sin(90-15) = \cos 15$$

Substituting these values into the expression, we get:

$$(\cos 75 - \sin 75)^2 = (\sin 15 - \cos 15)^2$$

Expanding the square, we get:

$$(\sin 15 - \cos 15)^2 = \sin^2 15 - 2 \sin 15 \cos 15 + \cos^2 15$$

Using another trigonometric identity, $\sin^2 x + \cos^2 x = 1$, we can simplify the expression further:

$$(\sin 15 - \cos 15)^2 = 1 - 2 \sin 15 \cos 15$$

To find the value of $\sin 15 \cos 15$, we can use the half-angle formula:

$$\sin 15 \cos 15 = \frac{1}{2} \sin 30 = \frac{1}{4}$$

Therefore, the final answer is:

$$(\cos 75 - \sin 75)^2 = 1 - 2 \sin 15 \cos 15 = 1 - \frac{1}{2} = \frac{1}{2}$$

I hope this helps you understand the solution. If you have any other math questions, feel free to ask me.Ответь+подробно.++Найдите+выражение+(cos+75+-+sin+75)+^2​+