Question

Решите cosx+sin^2x=cos^2x

Answer 1

Cosx+sin²x=cos²x
cosx+1-cos²x-cos²x=0
2cos²x-cosx-1=0
cosx=t
2t²-t-1=0
D=1+8=9=3²
t=(1±3)/4
t1=1;t2=-1/2
cosx=1;x=2πn
cosx=-1/2
x=±(π-π/3)+2πk
x=±2π/3+2πk;k€Z

Answer 2

To solve the trigonometric equation cos(x) + sin^2(x) = cos^2(x), we'll use some trigonometric identities to simplify the equation and find the solutions.

Step 1: Replace sin^2(x) with (1 - cos^2(x)) using the identity sin^2(x) + cos^2(x) = 1. cos(x) + (1 - cos^2(x)) = cos^2(x)

Step 2: Expand the equation: cos(x) + 1 - cos^2(x) = cos^2(x)

Step 3: Move all terms to one side of the equation: cos(x) + 1 = 2 * cos^2(x)

Step 4: Rewrite cos^2(x) as (1 - sin^2(x)) using the identity cos^2(x) + sin^2(x) = 1. cos(x) + 1 = 2 * (1 - sin^2(x))

Step 5: Move the term to the other side of the equation: 2 * sin^2(x) + cos(x) - 1 = 0

Now, we have a quadratic equation in terms of sin(x). Let's solve it.

Step 6: Set the equation to zero and factor if possible: 2 * sin^2(x) + cos(x) - 1 = 0

Step 7: Factor the quadratic equation: (2 * sin(x) - 1)(sin(x) + 1) = 0

Now, we have two possible factors that could result in the equation being true:

1. 2 * sin(x) - 1 = 0
2. sin(x) + 1 = 0

Let's solve for x in each case:

1. 2 * sin(x) - 1 = 0 2 * sin(x) = 1 sin(x) = 1/2

The solutions for sin(x) = 1/2 are x = π/6 and x = 5π/6. (Keep in mind that there are multiple solutions for trigonometric equations, as they are periodic.)

2. sin(x) + 1 = 0 sin(x) = -1

The solution for sin(x) = -1 is x = -π/2.

So, the solutions to the equation cos(x) + sin^2(x) = cos^2(x) are: x = π/6, 5π/6, -π/2, and other solutions that are periodic with a period of 2π.