Question

X = acoso+bsino, y = asino - bcoso then a^2+b^2=?​

Answer 1

Given

X=a cos∅+ b Sin∅

&

Y=a Sin∅-b cos∅

To find

a²+b²

Solution

Both side squaring

X²=(a cos∅+ b Sin∅)²

=>x²=a² cos²∅ + b² Sin²∅ + 2ab cos∅.sin∅__(1)

again both side squaring

Y²=(a Sin∅-b cos∅)²

=>y²=a² sin²∅ + b² cos²∅ - 2ab cos∅.sin∅__(2)

Joining Equation (1) and (2)

=>x² + y²=a² cos²∅ + b² Sin²∅ + 2ab cos∅.sin∅ + a² sin²∅ + b² cos²∅ - 2ab cos∅.sin∅

=>x²+y² = a²cos²∅ + a² sin²∅ + b² sin²∅+ b² cos²∅

=>x²+y²=a²(cos²∅ + sin²∅ )+ b²( sin²∅+ cos²∅)

=>x²+y²=a²+b²

hence,x²+y²=a²+b²