Sin75°+sin45°/sin285°

Sin 75° = sin(45 + 30) = sin45*cos30 + sin30*cos45 = (√2/2)* (√3/2) + (1/2)* (√2/2) = (√6/4) + (√2/4) = √2*(√3+1)/4.sin285° = (2*120 + 45) = sin(2*120)*cos45 + sin45*cos(2*120) = sin(2*120)* (√2/2) + (√2/2)*cos(2*120) = (√2/2)*(sin(2*120) + cos(2*120)) = (√2/2)*(2*sin120*cos120 + (cos120)^2 - (sin120)^2) = (√2/2)*(4*sin60*cos60*((cos60)^2 - (sin60)^2) + (cos120)^2 - (sin120)^2) == (√2/2)*(4*(√3/2)* (1/2)*((1/2)^2 - (√3/2)^2) + ((cos60)^2-(sin(60)^2)^2 - (2*sin60*cos60)^2) = = (√2/2)*( √3*(1/4 – ¾) + (¼ - ¾)^2 – (2*(√3/2)*(1/2))^2) = (√2/2)*( -2*√3/4 + 4/16 – ¾) == (√2/2)*( -2*√3/4 + 4/16 – ¾) = -(√2/2)*((2*4*√3+8)/16) = -√2*(√3+1)/4.Sin75°+sin45°/sin285° = √2*(√3+1)/4 + (√2/2)/ (-√2*(√3+1)/4) = √2*(√3+1)/4 + 4*√2/2*(-√2*(√3+1) = √2*(√3+1)/4 + 2*√2/(-√2*(√3+1) = (-2*(√3+1)*(√3+1) + 2√2)/ (-√2*(√3+1) == (-2*(3+2√3 + 1 + 2√2)/ (-√2*(√3+1) = (- 8 - 4√3 + 4√2)/ (-√2*(√3+1).