Question

Cosx+cos2x=2..............

Answer 1

Вот ответ
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Answer 2

To solve the equation cos(x) + cos(2x) = 2, you can follow these steps:

1. Start by applying the double-angle identity for cosine: cos(2x) = 2cos^2(x) - 1.

2. Substitute this expression into the original equation: cos(x) + (2cos^2(x) - 1) = 2.

3. Rearrange the equation to collect like terms: 2cos^2(x) + cos(x) - 3 = 0.

4. Now, you have a quadratic equation in terms of cos(x). To solve it, you can factor or use the quadratic formula. Let's use the quadratic formula: cos(x) = [-b ± √(b^2 - 4ac)] / (2a),

   where a = 2, b = 1, and c = -3.

5. Plug in these values into the formula: cos(x) = [-1 ± √(1^2 - 4(2)(-3))] / (2 * 2),

   cos(x) = [-1 ± √(1 + 24)] / 4,

   cos(x) = [-1 ± √25] / 4.

6. Simplify the expression further: cos(x) = [-1 ± 5] / 4.

7. Now, you have two possibilities: a) cos(x) = (5 - 1) / 4 = 4/4 = 1. b) cos(x) = (-1 - 5) / 4 = -6/4 = -3/2.

8. Since the cosine function's range is between -1 and 1, option b) is not possible, so you have: cos(x) = 1.

9. To find the solutions for x, take the arccosine (inverse cosine) of both sides: x = arccos(1).

10. The arccos(1) is equal to 0, so you have: x = 0.

So, the solution to the equation cos(x) + cos(2x) = 2 is x = 0.