Question

x= Acos +Bsin , y= Asin- Bcos prove that x²+y²=a²+b²

Answer 1

Answer:

QED

Step-by-step explanation:

X^2 + y^2

= (Acos + bsin)^2 + (asin - bcos)^2

= A^2 cos^2 + b^2sin^2 +2abcos sin + a^2sin^2 + b^2cos^2 - 2absincos

= A^2(sin^2+cos^2) +b^2(sin^2+cos^2)

As sin^2 + cos^2 = 1 so

= A^2 + b^2

QED

Answer 2

Answer:

Given :-

x = a cos ∅ - b sin ∅

Therefore,

x² = (a cos∅ - b sin∅)²  

x²= (a² Cos²∅) + (b² Sin2∅) - (2ab Cos∅ Sin∅)   ---------------( 1)

Also given ,

y = a sin ∅ + b cos ∅

Therefore,

y² = (a sin ∅  + b cos∅)y² = (a² Sin²∅) + (b² Cos²∅) + (2ab Cos∅  Sin∅)   ------(2)

Adding  (1 )and( 2),

x² + y² = (a² Cos²∅) + (b² Sin²∅)-(2ab Cos∅  Sin∅)+(a²Sin²∅) + (b² Cos²∅) +(2ab Cos∅  Sin∅)

Cancelling (2ab Cos∅  Sin∅) and - (2ab Cos∅  Sin∅)

 x² + y² = (a² Cos²∅) + (b²Sin²∅) + (a² Sin²∅) + (b² Cos²∅)

 

Bringing a² terms and b² terms together,

x² + y² = (a² Cos²∅) + (a² Sin²∅) + (b² Sin²∅) + (b² Cos²∅)

x² + y² = a² (Cos2∅  + Sin2∅) + b2 (Sin2∅  + Cos2∅)

By the identity Sin²∅  + Cos²∅  = 1

x² + y² = a2 (1) + b2 (1)

x² + y² = a² + b²

Hence, proved.