Question

If y = cot-1 (cosx)1/2 - tan-1(cosx)1/2 , prove that sin y = tan 2 (x/2)

Answer 1

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Answer 2

Answer:

Step-by-step explanation:

y=cot−1(cosx−−−−√)−tan−1(cosx−−−−√)   =tan−1(1cosx√)−tan−1(cosx−−−−√)   =tan−1(1cosx√−cosx√1+1cosx√×cosx√)   =tan−1(1−cosx2cosx√)⇒tany=1−cosx2cosx√⇒cot y=2cosx√1−cosxWe know thatcosec2y=1+cot2y               =1+(2cosx√1−cosx)2               =(1−cosx)2+4 cosx(1−cosx)2               =(1+cosx)2(1−cosx)2⇒cosec y=1+cosx1−cosx               =1+2cos2(x2)−11−1+2sin2(x2)               =cos2(x2)sin2(x2)y=cot-1cosx-tan-1cosx   =tan-11cosx-tan-1cosx   =tan-11cosx-cosx1+1cosx×cosx   =tan-11-cosx2cosx⇒tany=1-cosx2cosx⇒cot y=2cosx1-cosxWe know thatcosec2y=1+cot2y               =1+2cosx1-cosx2               =1-cosx2+4 cosx1-cosx2               =1+cosx21-cosx2⇒cosec y=1+cosx1-cosx               =1+2cos2x2-11-1+2sin2x2               =cos2x2sin2x2  

              =cot               =cot2x/2

⇒cosec y=cot⇒cosec y=cot2x/2

⇒1sin y=1tan2(x2)⇒sin y=tan 2(x/2)