Question

If 2 cos ¢ - sin¢ = X and cos ¢ - 3 sin ¢ = Y , prove that 2x²+y²-2xy = 5

Answer 1

Hii friend,

2 × Cos theta - Sin theta = X

And,


Cos theta - 3 Sin theta = Y


Therefore,


2X²+ Y² -2XY = 2(2Cos theta - sin theta)² + (Cos theta - 3 Sin theta)² - 2(2Cos theta + Sin theta)(Cos theta - 3 sin theta)

=> 2(4 × Cos² theta + Sin² theta - 4 × Cos theta × Sin theta) + Cos² theta+ 4 × Sin²theta - 6 × Sin theta × Cos theta - 2(2Cos² theta + 6 × Sin theta × Cos theta - Sin theta × Cos theta + 3 × Sin² theta )


=> 8 Cos²theta + 2 Sin²theta - 8 × Cos theta × Sin theta + Cos² theta + 9 × Sin²theta - 6 × Sin theta × Cos theta + 4 × Cos²theta + 12 × Sin theta × Cos theta + 2 × Sin theta × Cos theta - 6 × Sin theta




=> (5 ×Cos² theta + 5 × Sin²theta)

=> 5( Cos²theta + Sin²theta)


=> 5 × 1 = 5 = RHS


Hence,


LHS = RHS ......PROVED...



HOPE IT WILL HELP YOU....... :-)