Question

If cosec A + sec A=cosec B +sec B,prove that tan A+B/2=catA catB​

Answer 1

Answer:

Given : csc A + sec A = csc B + sec B ... (1) 

To Prove : ( tan A )( tan B ) = cot [ (A+B)/2 ] .. (2) 

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Proof : From (1), 

... sec A - sec B = csc B - csc A 

∴ (1/ cos A) - (1/ cos B) = (1/ sin B) - (1/ sin A) 

∴ ( cos B - cos A ) / ( cos A. cos B ) = ( sin A - sin B ) / ( sin A. sin B ) 

∴ rearranging the terms on two sides, 

... ( sin A / cos A )( sin B / cos B ) = ( sin A - sin B ) / ( cos B - cos B ) 

∴ ( tan A )( tan B ) = [ 2. cos (A+B)/2. sin (A-B)/2 ] / [ 2. sin ((A+B)/2. sin (A-B)/2 ] 

∴ ( tan A )( tan B ) = cot [ (A+B) / 2 ] Q.E.D.