Question

If x is equal to a sec theta + b tan theta and y is equal to a tan theta + b sec theta prove that x square minus y square is equal to a square minus b square

Answer 1

x² - y² = a² - b² if x = asecθ + btanθ  & y  = atanθ + bsecθ

Step-by-step explanation:

x = asecθ + btanθ

y  = atanθ + bsecθ

to be proved

x² - y² = a² - b²

LHS

= x² - y²

= (x + y)(x - y)

= (asecθ + btanθ + atanθ + bsecθ)(asecθ + btanθ - atanθ - bsecθ)

= (a(secθ + tanθ) + b(secθ + tanθ))(a(secθ - tanθ) - b(secθ - tanθ))

= (a + b)(secθ + tanθ)(a - b)(secθ - tanθ)

=  (a + b)(a - b)(secθ + tanθ)(secθ - tanθ)

= (a² - b²)(sec²θ - tan²θ)

=  (a² - b²)1

= a² - b²

= RHS

QED

Learn more:

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

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