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Exercise

sin(x+3y)+sin(3x+y)sin2x+sin2y\frac{sin\left(x+3y\right)+sin\left(3x+y\right)}{sin2x+sin2y}

Step-by-step Solution

1

Apply the trigonometric identity: sin⁡(a)+sin⁡(b)\sin\left(a\right)+\sin\left(b\right)=2sin⁡(a+b2)cos⁡(a−b2)=2\sin\left(\frac{a+b}{2}\right)\cos\left(\frac{a-b}{2}\right), where a=2xa=2x and b=2yb=2y

sin⁡(x+3y)+sin⁡(3x+y)2sin⁡(2x+2y2)cos⁡(2x−2y2)\frac{\sin\left(x+3y\right)+\sin\left(3x+y\right)}{2\sin\left(\frac{2x+2y}{2}\right)\cos\left(\frac{2x-2y}{2}\right)}

Final answer to the exercise

sin⁡(x+3y)+sin⁡(3x+y)2sin⁡(2x+2y2)cos⁡(2x−2y2)\frac{\sin\left(x+3y\right)+\sin\left(3x+y\right)}{2\sin\left(\frac{2x+2y}{2}\right)\cos\left(\frac{2x-2y}{2}\right)}

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