Question

1-2sin x cos x + sin x + cos x = 0

Answer 1

sinx+cosx=t  (sinx+cosx)=t²
sin²x+2sinxcosx+cos²x=1+2sinxcosx
2sinxcosx=t²-1
1-t²+1+t=0
t-t²=0
t(1-t)=0
t=0⇒sinx+cosx=0/cosx≠0
tgx+1=0⇒tgx=-1⇒x=-π/4+πn
t=1⇒sinx+cosx=1
2sinx/2cosx/2+cos²x/2-sin²x/2-sin²x/2-cos²x/2=0
2sinx/2cosx/2-2sin²x/2=0
2sinx/2(cosx/2-sinx/2)=0
sinx/2=0⇒x/2=πn⇒x=2πn
cosx/2-sinx/2=0/cosx/2≠0
1-tgx/4=0⇒tgx/4=1⇒x/4=π/4+πn⇒x=π+4πn
Ответ x=-π/4+πn; x=2πn; x=π+4πn  

Answer 2

To solve the equation 1 - 2sin(x)cos(x) + sin(x) + cos(x) = 0, we can manipulate it and use some trigonometric identities to simplify it.

Let's start by factoring out sin(x) from the first two terms and cos(x) from the last two terms:

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

Next, we can rewrite 1 + sin(x) as sin(x) + cos(x):

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

Now, let's expand the expression:

sin(x) - 2sin(x)cos(x) + cos(x)sin(x) + cos^2(x) = 0

Combine the middle two terms:

sin(x) - sin(x)cos(x) + cos^2(x) = 0

Rearrange the terms:

cos^2(x) - sin(x)cos(x) + sin(x) = 0

Now, let's use the trigonometric identity sin^2(x) + cos^2(x) = 1 to substitute cos^2(x) with 1 - sin^2(x):

(1 - sin^2(x)) - sin(x)cos(x) + sin(x) = 0

Expand the expression:

1 - sin^2(x) - sin(x)cos(x) + sin(x) = 0

Combine like terms:

1 - sin^2(x) - sin(x)cos(x) + sin(x) = 0

Rearrange the terms:

1 + sin(x) - sin^2(x) - sin(x)cos(x) = 0

Now, let's factor the expression:

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

Simplify further:

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

Now, we have two factors: (1 - sin(x)) = 0 and (1 + sin(x) - cos(x)) = 0.

Setting (1 - sin(x)) = 0 gives sin(x) = 1. Solving this equation, we find that x = π/2 + 2πn, where n is an integer.

Setting (1 + sin(x) - cos(x)) = 0 is a bit more complicated. There is no simple analytical solution for this equation, so you would need to use numerical methods or approximation techniques to find the values of x that satisfy it.

Therefore, the solutions to the original equation 1 - 2sin(x)cos(x) + sin(x) + cos(x) = 0 are:

x = π/2 + 2πn, where n is an integer, and the solutions of the equation (1 + sin(x) - cos(x)) = 0 obtained through numerical methods or approximations.