If x = a secθ + b tanθ and y = a tanθ + b secθ, then prove that x² – y² = a² – b².

x² =(a secθ + b tanθ)²

=a²sec²θ + b²tan²θ + ab secθ tanθ

y² =(a tanθ + b secθ)²

=a²tan²θ + b²sec²θ + ab secθ tanθ

Now,

LHS = x²– y²

= (a²sec²θ + b²tan²θ + ab secθ tanθ)

- (a²tan²θ + b²sec²θ + ab secθ tanθ)

= a²sec²θ + b²tan²θ + ab secθ tanθ

- a²tan²θ - b²sec²θ - ab secθ tanθ

= a² (sec²θ –tan²θ) – b² (sec²θ –tan²θ)

=a² – b² [∵ sec²θ –tan²θ =1]

=RHS

Hence proved.