Question

Sin (π/4-α) * cos (π/4-α)

Answer 1

Відповідь:

Расскладываем синус и косинус по формулам суммы и разности.

(sin(pi/4-a)+cos(pi/4-a))/(sin(pi/4-a)-cos(pi/4-a))=

(sin(pi/4)*cosa-cos(pi/4)*sina+cos(pi/4)*cosa-sina*sin(pi/4))/(sin(pi/4)*cosa-cos(pi/4)*sina-cos(pi/4)*cosa+sina*sin(pi/4))=(здесь сокращаем все синусы и косинусы угла пи на 4, так как они равны и получаем) = (cosa-sina+cosa+sina)/cosa-sina-cosa-sina)=2cosa/(-2sina)=-ctga

Покрокове пояснення:

Answer 2

To simplify the expression, let's use the trigonometric identities:

1. sin(a - b) = sin(a) * cos(b) - cos(a) * sin(b)
2. cos(a - b) = cos(a) * cos(b) + sin(a) * sin(b)

Applying these identities to the given expression:

sin(π/4 - α) * cos(π/4 - α)

= sin(π/4) * cos(α) - cos(π/4) * sin(α) * cos(α) (Applying identity 1) = (1/√2) * cos(α) - (1/√2) * sin(α) * cos(α) (Simplifying sin(π/4) and cos(π/4) as 1/√2) = (1/√2) * cos(α) - (1/√2) * cos(α) * sin(α) (Rearranging terms)

Now, you can factor out a common term of cos(α):

= cos(α) * ((1/√2) - (1/√2) * sin(α)) = cos(α) * (1/√2) * (1 - sin(α))

So, the simplified expression is cos(α) * (1/√2) * (1 - sin(α)).