Question

Cos 5x × cosx =cos 4x​

Answer 1

Ответ:

1/2*(cos(5x-x)+cos(5x+x))-cos4x=0

1/2*cos4x+1/2*cos6x-cos4x=0

1/2*cos6x-1/2*cos4x=0

1/2*(cos6x-sin4x)=0

1/2*(-2sin5xsinx)=0

sin5x=0⇒5x=πn,n∈z⇒x=πn/5,n∈z

sinx=0⇒x=πk,k∈z

Ответ x=πn/5,n∈z

Пошаговое объяснение:

Answer 2

To solve the equation cos(5x) * cos(x) = cos(4x), we can use trigonometric identities and simplification techniques. Let's go step by step:

1. Start with the left side of the equation: cos(5x) * cos(x).

2. Apply the double angle identity for cosine: cos(2θ) = 2cos²(θ) - 1.

3. Rewrite cos(5x) as cos(4x + x) and cos(4x) as cos(2(2x)).

4. Apply the double angle identity to cos(2(2x)) to get cos(4x): cos(2(2x)) = 2cos²(2x) - 1.

5. Replace cos(4x) in the original equation with the expression from step 4: cos(5x) * cos(x) = (2cos²(2x) - 1) * cos(x).

6. Distribute the cos(x) term: cos(5x) * cos(x) = 2cos²(2x)cos(x) - cos(x).

7. Apply the double angle identity to cos(2x) to get cos²(2x): cos²(2x) = (1 + cos(4x)) / 2.

8. Substitute cos²(2x) in the equation from step 6: cos(5x) * cos(x) = 2(1 + cos(4x)) / 2 * cos(x) - cos(x).

9. Simplify the expression: cos(5x) * cos(x) = (1 + cos(4x)) * cos(x) - cos(x).

10. Expand the equation: cos(5x) * cos(x) = cos(x) + cos(4x) * cos(x) - cos(x).

11. Cancel out the common terms: cos(5x) * cos(x) = cos(4x) * cos(x).

12. Divide both sides by cos(x): cos(5x) = cos(4x).

13. Apply the identity cos(a) = cos(b): 5x = 4x.

14. Subtract 4x from both sides: 5x - 4x = 0.

15. Simplify: x = 0.

Therefore, the solution to the equation cos(5x) * cos(x) = cos(4x) is x = 0.