Question

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

Answer 1

sinx + sin^2 x/2 = cos^2 x/2

sinx = cos^2 x/2- sin^2 x/2

 

чёрным выделен фрагмент, который можно свернуть в косинус двойного аргумента

cos^2 x/2- sin^2 x/2= cos2*(x/2)=cosx

 

перепишем

sin x = cos x

tg x= 1

x= п/4+ пn

 

Answer 2

To solve the equation sin(x) + sin^2(x/2) = cos^2(x/2), we can use the trigonometric identity sin^2(x) + cos^2(x) = 1 to get rid of the squared terms:

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

Simplifying this equation, we get:

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

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

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

Taking the square root of both sides, we get:

cos(x/2) = ±sqrt((1 + sin(x))/2)

Squaring both sides, we get:

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

Substituting this into the original equation, we get:

sin(x) + (1 + sin(x))/2 = (1 + sin(x))/2

Simplifying this equation, we get:

sin(x) = 0

Therefore, the solutions to the equation sin(x) + sin^2(x/2) = cos^2(x/2) are all values of x for which sin(x) = 0, i.e., x = kπ, where k is an integer.