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Question

If x = a sec θ + b tan θ and y = a tan θ + b sec θ, prove that (x2 − y2) = (a2 − b2).

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Solution

We have x2y2=[asecθ+btanθ2-atanθ+bsecθ2] = (a2sec2θ+b2tan2θ+2absecθtanθ)(a2tan2θ+b2sec2θ+2abtanθsecθ) = a2sec2θ+b2tan2θa2tan2θb2sec2θ =(a2sec2θa2tan2θ)(b2sec2θb2tan2θ) =a2(sec2θtan2θ)b2(sec2θtan2θ) =a2b2 [sec2θtan2θ=1]Hence, x2y2=a2b2

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