Question

Cos2п/15 cosp/5+sin2p/15 sinp/5

Answer 1

Ответ:

Пошаговое объяснение:     Зная, что

cos(α – β) = cos α · cos β + sin α · sin β и    cos(-a)=cosa

cos2п/15 cosp/5+sin2p/15 sinp/5=cos(2п/15-п/5)=cos(2п/15-3п/15)=

=cos(-п/15)=cosп/15

Answer 2

To simplify the given expression, let's break it down step by step:

1. Recall the trigonometric identity: cos(2θ) = 1 - 2sin²(θ). We can rewrite cos²(θ) as (1 - sin²(θ)).

2. Substituting θ = π/15 into the identity, we have: cos(2π/15) = 1 - 2sin²(π/15).

3. Similarly, we can rewrite cos(π/5) as (1 - sin²(π/5)).

4. Now, let's simplify the given expression: cos²(π/15) cos(π/5) + sin²(π/15) sin(π/5).

   Substituting the expressions from steps 2 and 3, we have: (1 - 2sin²(π/15))(1 - sin²(π/5)) + sin²(π/15) sin(π/5).

5. Expand the expression: (1 - 2sin²(π/15))(1 - sin²(π/5)) + sin²(π/15) sin(π/5) = 1 - sin²(π/5) - 2sin²(π/15) + 2sin²(π/15) sin²(π/5) + sin²(π/15) sin(π/5).

6. Notice that sin²(π/15) appears in all terms. Let's factor it out: sin²(π/15)(1 - 2sin²(π/5) + 2sin²(π/5) sin²(π/15) + sin(π/5)).

7. Now, let's simplify the expression inside the parentheses: 1 - 2sin²(π/5) + 2sin²(π/5) sin²(π/15) + sin(π/5).

8. We can rewrite sin(π/5) as √(1 - cos²(π/5)). Let's substitute that in: 1 - 2sin²(π/5) + 2sin²(π/5) sin²(π/15) + √(1 - cos²(π/5)).

At this point, we have simplified the given expression as much as possible.