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Answered

if m=a sec A+b tan A and n=a tan A+b sec A,then prove that:m square-n square=a square-b square

Answer :

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m²-n²=(m+n)(m-n)    
   ={(a sec A+ b tan A)+ (a tan A+ b sec A)}{(a sec A+ b tan A)-(a tan A+ b sec A)}
   ={(a+b) sec A+(a+b) tan A}*{(a-b) sec A- (a-b) tan A}
   =(a+b)(sec A+ tan A)*(a-b) (sec A -tan A)
   = (a+b) (a-b) (sec A +tan A)(sec A- tan A)
   =(a²-b²)(sec²A-tan²A)
   =(a²-b²)(1+tan²A-tan²A)
   =a² - b²

LHS = tan²A - sin²A

= ( sin²A/cos²A ) - sin²A

= sin²A [ 1/cos²A - 1 ]

= sin²A( sec²A - 1 )

= sin²A tan²A

[ since , sec²A - 1 = tan²A ]

= tan²Asin²A

= RHS

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