Question

Помогите разобраться: sin^2 П/8 * cos^2 П/8 = 2(sin П/8 * cos П/8)^2/2 = (2*sin П/8* cos П/8)^2/4 = (sin 2* П/8)^2/4=... Почему число 2 в знаменателе стало 4?

Answer 1

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Answer 2

Потому что 2 в скобки включили, где квадрат =>
${2}^{2} = 4$

Answer 3

Calculation Explanation

To understand why the number 2 in the denominator became 4, let's break down the given expression step by step:

1. sin^2(π/8) * cos^2(π/8) - This expression represents the product of the squares of the sine and cosine of π/8.

2. 2(sin(π/8) * cos(π/8))^2 / 2 - Here, we have multiplied the expression from step 1 by 2 and then divided it by 2. This step does not change the value of the expression.

3. (2 * sin(π/8) * cos(π/8))^2 / 4 - In this step, we have simplified the expression by factoring out a 2 from the numerator and denominator. This does not change the value of the expression.

4. (sin(2 * π/8))^2 / 4 - Finally, we use the double-angle identity for sine, which states that sin(2θ) = 2sin(θ)cos(θ). In this case, θ is π/8. So, sin(2 * π/8) = 2sin(π/8)cos(π/8). Substituting this into the expression gives us (sin(2 * π/8))^2 / 4.

Therefore, the expression simplifies to (sin(2 * π/8))^2 / 4.

Simplification Explanation

Now, let's simplify the expression further using the double-angle identity for sine:

1. sin(2 * π/8) - Using the double-angle identity for sine, sin(2θ) = 2sin(θ)cos(θ), we can rewrite sin(2 * π/8) as 2sin(π/8)cos(π/8).

2. (2sin(π/8)cos(π/8))^2 / 4 - Substituting the value of sin(2 * π/8) from step 1 into the expression, we get (2sin(π/8)cos(π/8))^2 / 4.

Therefore, the simplified expression is (2sin(π/8)cos(π/8))^2 / 4.

Conclusion

To summarize, the expression sin^2(π/8) * cos^2(π/8) simplifies to (2sin(π/8)cos(π/8))^2 / 4. The number 2 in the denominator became 4 after factoring out a 2 from the numerator and denominator.