1.

X=asectheta+btantheta y=atantheta+bsectheta prove that x2-y2=a2-b2​

Answer»

Solution:-

Formula Used:-

  • (a + b)² = a² + 2ab + b²

  • sec²θ - tan²θ = 1

  • tan²θ - sec²θ = -1

Given:-

▪︎x = asecθ + btanθ

By SQUARING both SIDES:

➟ (x)² = (asecθ + btanθ)²

➟ x² = a²sec²θ + 2asecθbtanθ + b²tan²θ ------ eq.1

And

▪︎y = atanθ + bsecθ

By squaring both sides:

➟ (y)² = (atanθ + bsecθ)²

➟ y² = a²tan²θ + 2atanθsecθ + b²sec²θ ------ eq.2

Now,

L.H.S. = x² - y²

Putting eq.1 and eq.2,

= a²sec²θ + 2asecθbtanθ + b²tan²θ - (a²tan²θ + 2atanθsecθ + b²sec²θ)

= a²sec²θ + 2asecθbtanθ + b²tan²θ - a²tan²θ - 2atanθsecθ - b²sec²θ

= a²sec²θ - a²tan²θ + b²tan²θ - b²sec²θ

= a²(sec²θ - tan²θ) + b²(tan²θ - sec²θ)

= a² × 1 + b² × -1

= a² - b²

= R.H.S.

Hence, PROVED.


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