CBSE Class 10 Answered
Prove that-
tan A/1-cot A + cot A/1-tan A=1+tan A+cot A
Asked by Aditya Chakravarty | 23 May, 2013, 12:44: PM
Expert Answer
Answer : To prove : tan A/1-cot A + cot A/1-tan A=1+tan A+cot A
taking L.H.S
= [ tan(A) / (1 - cot(A)) ] + [ cot(A) / (1 - tan(A) ) ]
= [ sin(A)/cos(A) / (1 - cos(A)/sin(A)) ] + [ cos(A)/sin(A) / (1 - sin(A)/cos(A) ) ]
= [ sin(A)/cos(A) / (1 - cos(A)/sin(A)) ] + [ cos(A)/sin(A) / (1 - sin(A)/cos(A) ) ]
{using tan x = sinx /cos x and cotx = cosx/sinx }
= [ sin(A)/cos(A) / (sin(A)/sin(A) - cos(A)/sin(A)) ] + [ cos(A)/sin(A) / (cos(A)/cos(A) - sin(A)/cos(A) ) ]
= [ sin(A)/cos(A) / (sin(A) - cos(A)) / sin(A) ] + [ ( cos(A)/sin(A) / ( cos(A) -sin(A) ) / cos(A) ) ]
= [ sin(A)sin(A) / cos(A)(sin(A) - cos(A)) ] + [ ( cos(A)cos(A) / sin(A)( cos(A) -sin(A) ) ]
=[ sin2(A) / cos(A)(sin(A) - cos(A)) ] + [ cos2(A) / -sin(A)( sin(A) - cos(A) ) ) ]
=[ sin2(A) / cos(A)(sin(A) - cos(A)) ] - [ cos2(A) / sin(A)( sin(A) - cos(A) ) ) ] ===> LCD
= [ sin2(A) sin(A) / cos(A)sin(A)(sin(A) - cos(A)) ] - [ cos2(A) cos(A) / sin(A)cos(A)( sin(A) - cos(A) ) ) ]
= [ sin3(A) / cos(A)sin(A)(sin(A) - cos(A)) ] - [ cos3(A) / sin(A)cos(A)( sin(A) - cos(A) ) ) ]
=[ sin3(A) - cos3(A) ] / [ sin(A)cos(A)( sin(A) - cos(A) ) ) ]
=[ (sin(A) - cos(A)) ( sin2(A) + sin(A) cos(A) + cos2(A) ) ] / [ sin(A)cos(A)( sin(A) - cos(A) ) ) ]
= [ ( sin2(A) + cos2(A) + sin(A) cos(A) ] / [ sin(A)cos(A) ]
= [ sin2(A) / sin(A)cos(A) ] + [ cos2(A) / sin(A)cos(A) ] + [ sin(A) cos(A) / sin(A) cos(A) ]
= [ sin(A) / cos(A) ] + [ cos(A) / sin(A) ] + 1
= tan(A) + cot(A) + 1
= [ sin(A)/cos(A) / (sin(A)/sin(A) - cos(A)/sin(A)) ] + [ cos(A)/sin(A) / (cos(A)/cos(A) - sin(A)/cos(A) ) ]
= [ sin(A)/cos(A) / (sin(A) - cos(A)) / sin(A) ] + [ ( cos(A)/sin(A) / ( cos(A) -sin(A) ) / cos(A) ) ]
= [ sin(A)sin(A) / cos(A)(sin(A) - cos(A)) ] + [ ( cos(A)cos(A) / sin(A)( cos(A) -sin(A) ) ]
=[ sin2(A) / cos(A)(sin(A) - cos(A)) ] + [ cos2(A) / -sin(A)( sin(A) - cos(A) ) ) ]
=[ sin2(A) / cos(A)(sin(A) - cos(A)) ] - [ cos2(A) / sin(A)( sin(A) - cos(A) ) ) ] ===> LCD
= [ sin2(A) sin(A) / cos(A)sin(A)(sin(A) - cos(A)) ] - [ cos2(A) cos(A) / sin(A)cos(A)( sin(A) - cos(A) ) ) ]
= [ sin3(A) / cos(A)sin(A)(sin(A) - cos(A)) ] - [ cos3(A) / sin(A)cos(A)( sin(A) - cos(A) ) ) ]
=[ sin3(A) - cos3(A) ] / [ sin(A)cos(A)( sin(A) - cos(A) ) ) ]
=[ (sin(A) - cos(A)) ( sin2(A) + sin(A) cos(A) + cos2(A) ) ] / [ sin(A)cos(A)( sin(A) - cos(A) ) ) ]
= [ ( sin2(A) + cos2(A) + sin(A) cos(A) ] / [ sin(A)cos(A) ]
= [ sin2(A) / sin(A)cos(A) ] + [ cos2(A) / sin(A)cos(A) ] + [ sin(A) cos(A) / sin(A) cos(A) ]
= [ sin(A) / cos(A) ] + [ cos(A) / sin(A) ] + 1
= tan(A) + cot(A) + 1
{we know sin x/ cos x = tan x and cos x/sin x = cot x }
= R. H . S
Hence proved
Answered by | 23 May, 2013, 08:06: PM
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