Question

2sin^2 2x=(cosx+sinx)^2​

Answer 1

2sin^2 2x=(cosx+sinx)^2​

2(2sinxcox)^2 =cos^2(x)+2sinxcosx+sin^2(x)

2(2sinxcox)^2 =2sinxcosx+1

2(2sinxcox)^2-2sinxcosx-1=0

сделаем замену

2sinxcosx=sin2x=t

2t^2-t-1=0

D=(-1)^2-4*2*(-1)=1+8=9

√D=√9=3

t1=(1+3)/2*2=1

t2=(1-3)/4=-2/4=-1/2

делаем обратную замену

sin2x=1      

2х=π/2+2πn. n∈Z

x=π/4+πn, n∈Z

sin2x= -1/2

2x= (-1)^k arcsin(-1/2)+πk, k∈Z

2x=(-1)^k+1 π/6+πk, k∈Z

x=(-1)^k+1 π/12+πk/2, k∈Z

Ответ:

x1=π/4+πn, n∈Z

x2=(-1)^k+1 π/12+πk/2, k∈Z

Answer 2

To solve the equation 2sin^2(2x) = (cos(x) + sin(x))^2, we can use trigonometric identities to simplify and manipulate the equation.

Let's start by expanding the right side of the equation using the identity (a + b)^2 = a^2 + 2ab + b^2:

2sin^2(2x) = (cos(x) + sin(x))^2 2sin^2(2x) = cos^2(x) + 2cos(x)sin(x) + sin^2(x)

Next, we can use the Pythagorean identity sin^2(x) + cos^2(x) = 1 to simplify the equation further:

2sin^2(2x) = 1 + 2cos(x)sin(x) + sin^2(x)

Since sin^2(2x) can be written as 1/2 - 1/2cos(4x) using the double-angle identity for sine, the equation becomes:

1/2 - 1/2cos(4x) = 1 + 2cos(x)sin(x) + sin^2(x)

Now, let's combine like terms and rearrange the equation:

-1/2cos(4x) = 1 + 2cos(x)sin(x) + sin^2(x) - 1/2

-1/2cos(4x) = 1/2 + 2cos(x)sin(x) + sin^2(x)

Now, let's simplify the equation further by using the double-angle identity for cosine:

-1/2[cos^2(2x) - sin^2(2x)] = 1/2 + 2cos(x)sin(x) + sin^2(x)

-1/2cos^2(2x) + 1/2sin^2(2x) = 1/2 + 2cos(x)sin(x) + sin^2(x)

Now, let's substitute u = sin(2x):

-1/2(1 - u^2) + 1/2u^2 = 1/2 + 2cos(x)sin(x) + sin^2(x)

-1/2 + 1/2u^2 + 1/2u^2 = 1/2 + 2cos(x)sin(x) + sin^2(x)

1/2u^2 + 1/2u^2 = 1/2 + 2cos(x)sin(x) + sin^2(x) + 1/2

u^2 + u^2 = 1 + 4cos(x)sin(x) + 2sin^2(x) + 1

2u^2 = 2 + 4cos(x)sin(x) + 2sin^2(x)

Simplifying further:

u^2 = 1 + 2cos(x)sin(x) + sin^2(x)

Substituting u = sin(2x) back into the equation:

sin^2(2x) = 1 + 2cos(x)sin(x) + sin^2(x)

Now we have an equation in terms of sin(2x). To solve for x, we need to use trigonometric identities and techniques such as factoring, completing the square, or the quadratic formula to simplify and solve the equation.