Question

Cos 2x cos x – sin 2x sin x =0

Answer 1

cos 2x cos x – sin 2x sin x =0

cos(2x+x)=0

cos3x=0

3x=П/2+Пn

x=П/6+Пn/3

Ответ:П/6+Пn/3

Answer 2

To solve the equation cos(2x)cos(x) - sin(2x)sin(x) = 0, we can use the trigonometric identity:

cos(2θ) = cos²(θ) - sin²(θ)

Using this identity, we can rewrite the equation as:

cos²(x) - sin²(x) - sin(2x)sin(x) = 0

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

sin(2θ) = 2sin(θ)cos(θ)

The equation becomes:

cos²(x) - sin²(x) - 2sin²(x)cos(x) = 0

Using the Pythagorean identity: cos²(x) = 1 - sin²(x), we can substitute it in:

1 - sin²(x) - sin²(x) - 2sin²(x)cos(x) = 0

Rearranging the terms:

1 - 4sin²(x)cos(x) = 0

Factoring out sin(x):

sin(x)(1 - 4sin(x)cos(x)) = 0

Now, we have two possibilities for the equation to be true:

1. sin(x) = 0

This occurs when x is an integer multiple of π:

x = nπ, where n is an integer.

2. 1 - 4sin(x)cos(x) = 0

Dividing both sides by cos(x):

(1/cos(x)) - 4sin(x) = 0

Using the identity: sec(x) = 1/cos(x):

sec(x) - 4sin(x) = 0

Dividing through by sec(x):

1 - 4sin(x)tan(x) = 0

Now, we have a quadratic equation in terms of tan(x):

4tan²(x) - tan(x) + 1 = 0

Using the quadratic formula:

tan(x) = [ -b ± sqrt(b² - 4ac) ] / 2a

tan(x) = [ -(-1) ± sqrt((-1)² - 4(4)(1)) ] / (2(4))

tan(x) = [ 1 ± sqrt(1 - 16) ] / 8

tan(x) = [ 1 ± sqrt(-15) ] / 8

Since the square root of a negative number is undefined in the real number system, there are no real solutions for the quadratic equation.

Therefore, the solutions to the given equation are:

x = nπ, where n is an integer.