Question

Ques-30
if x= a sec theta + b tan theta and y= a tan theta + b sec theta, proof that x²-y²=a²-b².
pls anyone solve fast ⏩⏩​

Answer 1

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→ 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.

Answer 2

→ 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.