solve for x: sin(x+π/4)+sin(x-π/4)=-1

Answer 1

Use the identity sin(a±b)=sin(a)cos(b)±sin(b)cos(a)

Thus sin(x±π/4)=sin(x)cos(π/4)±sin(π/4)cos(x) = √2/2(sinx±cosx)

so sin(x+π/4)+sin(x-π/4)=√2/2(sinx+cosx)+√2/2(sinx-cosx) = √2/2(sinx+cosx+sinx-cosx) = √2sinx

so √2sinx=-1 => sinx = -1/√2 = -√2/2 => x =-π/4+2nπ and x=-3π/4+2nπ where n is any integer.