Question Number 149598 by naka3546 last updated on 06/Aug/21

Suppose  that    sec x + tan x = ((22)/7)  cosec x + cot x = (m/n)  (m/n)  is  in  the  lowest  term .  Find  m + n .

Answered by iloveisrael last updated on 06/Aug/21

 ((1+sin x)/(cos x))=((22)/7) ⇒((1−sin ^2 x)/(cos x(1−sin x)))=((22)/7)  ⇒((cos x)/(1−sin x))=((22)/7)  ⇒((cos ^2 (x/2)−sin^2 (x/2) )/((cos (x/2)−sin (x/2))^2 ))=((22)/7)  ⇒((cos (x/2)+sin (x/2))/(cos (x/2)−sin (x/2)))=((22)/7)  ⇒7cos (x/2)+7sin (x/2)=22cos (x/2)−22sin (x/2)  ⇒29 sin (x/2)=15 cos (x/2)  ⇒tan (x/2)=((15)/(29))  ⇒((1+cos x)/(sin x))=(m/n)  ⇒((2cos ^2 (x/2))/(2sin (x/2)cos (x/2)))=(m/n)  ⇒cot (x/2)=(m/n)=((29)/(15))  ⇒m+n=44

Answered by nimnim last updated on 06/Aug/21

         secx+tanx=((22)/7).....(i)  ⇒   secx−tanx=(7/(22))....(ii)  (i)+(ii)⇒2secx=(((22)/7)+(7/(22)))⇒secx=((533)/(308))  (i)−(ii)=2tanx=(((22)/7)−(7/(22)))⇒tanx=((435)/(308))         cosecx=((secx)/(tanx))=((533)/(435))  and  cotx=((308)/(435))        ⇒ cosecx+cotx=((533)/(435))+((308)/(435))=(m/n)          ⇒((841)/(435))=(m/n)⇒((29)/(15))=(m/n)      ∴ m+n=29+15=44★