If X = a Sec θ + B Tan θ and Y = a Tan θ + B Sec θ Prove that X2 - Y2 = A2 - B2. - Mathematics

If x = a sec θ + b tan θ and y = a tan θ + b sec θ prove that x² - y² = a²- b².

Sum

Here,
x²= a² sec²θ + 2ab sec θ.tan θ + b²tan²θ
y²= a²tan²θ + 2ab sec θ.tan θ + b²sec²θ

⇒ x²- y² = a² ( sec²θ - tan²θ ) - b² ( sec²θ - tan²θ )

⇒ x² - y² = a²- b². ....( ∵ sec²θ - tan²θ = 1)
Hence proved.

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