sin (x/2)=2 sin (x/4)cos(x/4) cos(x/2)=cos²(x/4)-sin²(x/4) 1=sin²(x/4)+cos²(x/4)
Уравнение примет вид: 2 sin (x/4)cos(x/4)-3·(cos²(x/4)-sin²(x/4))=3·(sin²(x/4)+cos²(x/4)) или 2 sin (x/4)cos(x/4)-3·cos²(x/4)+ 3·sin²(x/4)=3·sin²(x/4)+ 3·cos²(x/4)
2 sin (x/4)cos(x/4)-6·cos²(x/4)=0
2·cos(x/4)·(sin(x/4)-3cos(x/4))=0
cos(x/4)=0 или sin(x/4)-3cos(x/4)=0
х/4=π/2 + πk, k∈ Z или tg(x/4)=3 x=2π+4πk,k∈Z x/4=arctg 3 + πn, n∈Z x=4arctg 3 + 4πn, n∈Z
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Verified answer
Применим формулу половинного аргументаsin (x/2)=2 sin (x/4)cos(x/4)
cos(x/2)=cos²(x/4)-sin²(x/4)
1=sin²(x/4)+cos²(x/4)
Уравнение примет вид:
2 sin (x/4)cos(x/4)-3·(cos²(x/4)-sin²(x/4))=3·(sin²(x/4)+cos²(x/4))
или
2 sin (x/4)cos(x/4)-3·cos²(x/4)+ 3·sin²(x/4)=3·sin²(x/4)+ 3·cos²(x/4)
2 sin (x/4)cos(x/4)-6·cos²(x/4)=0
2·cos(x/4)·(sin(x/4)-3cos(x/4))=0
cos(x/4)=0 или sin(x/4)-3cos(x/4)=0
х/4=π/2 + πk, k∈ Z или tg(x/4)=3
x=2π+4πk,k∈Z x/4=arctg 3 + πn, n∈Z
x=4arctg 3 + 4πn, n∈Z
Ответ. x=2π + 4πk,k∈Z ; x=4arctg 3 + 4πn, n∈Z