Question

If x=a sec theta+b tan theta ,y=a tan theta +b sec theta prove that x 2 -y ​2 =a 2 -b 2

Answer 1

let A = theta

x² - y² = a² sec² A + b² tan² A + 2 ab secA tan A
                  - (a² tan² A + b² sec²A + 2ab tanA secA)
          = a² (sec²A - tan²A) + b² (sec²A - tan²A)
          = a² - b²

Answer 2

Answer :-

→ x² - y² = a² - b² .

Step-by-step explanation :-

We have,

x = a sec ∅ + b tan ∅ .

y = a tan ∅ + b sec ∅ .

°•° x² –y² .

= (a sec ∅ + b tan ∅)² – (a tan ∅ + b sec ∅ )² .

= ( a² sec²∅ + b²tan²∅ + 2 ab sec ∅. tan∅ ) – (a² tan²∅ + b² sec²∅ + 2ab sec ∅. tan∅ ) .

= a² sec²∅ + b² tan²∅+ 2ab sec∅. tan∅ – a²tan²∅ – b²sec²∅ – 2ab sec ∅.tan∅

= a² (sec²∅ – tan²∅ ) + b²(tan²∅ – sec²∅ ) .

= a² – b² .

Hence, it is proved.