Question

solve the trigonometry question for me​

Answer 1

Given--->

x Sinθ / a - y Cosθ / b = 1

x Cosθ / a + ySinθ / b = 1

To prove ----> x² / a² + y² / b² = 1

Proof---> ATQ,

x Sinθ / a - yCosθ / b = 1 ...............(1)

x Cosθ / a + y Sinθ / b = 1 ................(2)

Squaring equation(1) and (2) and then adding

( xSinθ/a - yCosθ/b )² + ( xCosθ/a + ySinθ/b)²

= 1² + 1²

We have an identiy, as follows

( a + b )² = a² + b² + 2ab, applying it here ,we get,

=> x²Sin²θ/a² + y²Cos²θ/b² - 2xySinθ Cosθ / ab

+ x²Cos²θ/a² + y²Sin²θ/b²+2xySinθCosθ/ab =2

=> x²/a² (Sin²θ +Cos²θ) + y²/b² ( Sin²θ+Cos²θ )= 2

We know that , Sin²A + Cos²A = 1 , applying it here , we get,

=> x² / a² ( 1 ) + y² / b² ( 1 ) = 2

=> x² / a² + y² / b² = 2