Question

If cosec A + sec A = cosec B + sec B then prove that tanA tanB = cot A+B/2

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Answer 1

ANSWER:

GIVEN:

cosec A +sec A = cosec B +sec B

TO PROOVE:

tan A.tan B= cot (A+B /2)

PROOF:

cosec A+ sec A = cosec B + sec B

=> 1/sin A + 1/ cos A = 1/ cos B + 1/ sin B

=>cos A + sin A/ sin A cos A = cos B + sin B / sin B cos B

=> sin A sin B cos B + cos B sin A cos A + sin A sin B cos A

=> sin A sin B cos A - sin A sin B cos B = cos A cos B sin B - cos A cos B sin A

=> sin A sin B ( cos A - Cos B) = cos A cos B ( sin B - sin A )

=> sin A sin B [ 2 sin (A+B /2) sin (B- A /2) ] = cos A cos B [ 2 cos (A+B/2) sin (B-A/2) ]

=> sin A sin B/ cos A cos B = cos (A+B/2)/ sin (A+B/2)

=> tan A tan B = cot (A+B/2)

hence proved.....

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