Q52 of 268 Page 5

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

Taking LHS =x2 – y2

Putting the values of x and y, we get


(a sec θ + b tan θ)2 – (a tan θ + b sec θ)2


= a2 sec2θ + b2 tan2θ + 2ab sec θ tan θ – a2tan2θ – b2 sec2θ – 2ab sec θ tan θ


= a2 (sec2θ – tan2θ) – b2 (sec2θ – tan2θ)


= a2 – b2 [∵ 1+ tan2 θ = sec2 θ]


=RHS


Hence Proved


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