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If-for-0-lt-x-lt-pi-2-y-exp-sin-2-x-sin-4-x-sin-6-x-log-e-2-is-a-zero-of-the-quadratic-equation-x-2-9x-8-0-then-the-value-of-sin-x-cos-x-sin-x-cos-x-is-




Question Number 56415 by gunawan last updated on 16/Mar/19
If   for  0<x<(π/2) ,   y=exp[(sin^2 x+sin^4 x+sin^6 x+...∞)log_e 2]  is a zero of the quadratic equation   x^2 −9x+8=0, then the value of((sin x+cos x)/(sin x−cos x)) is
Iffor0<x<π2,y=exp[(sin2x+sin4x+sin6x+)loge2]isazeroofthequadraticequationx29x+8=0,thenthevalueofsinx+cosxsinxcosxis
Answered by tanmay.chaudhury50@gmail.com last updated on 16/Mar/19
y=e^([(sin^2 x+sin^4 x+...∞)ln2])   now s=sin^2 x+sin^4 x+...∞  s=((sin^2 x)/(1−sin^2 x))=tan^2 x  y=e^(tan^2 x×ln2)   x^2 −9x+8=0  (x−1)(x−8)=0  so value of y is either=1   or  8  1=e^(tan^2 x×ln2)   e^0 =e^(tan^2 x×ln2)   but tanx≠0  so 8=e^(tan^2 x×ln2)   tan^2 x×ln2=ln8  tan^2 x×ln2=3ln2  tanx=(√3)   ((sinx+cosx)/(sinx−cosx))  =((tanx+1)/(tanx−1))→(((√3) +1)/( (√3) −1))  ((3+2(√3) +1)/(3−1))  =((4+2(√3))/2)  =2+(√3) answer
y=e[(sin2x+sin4x+)ln2]nows=sin2x+sin4x+s=sin2x1sin2x=tan2xy=etan2x×ln2x29x+8=0(x1)(x8)=0sovalueofyiseither=1or81=etan2x×ln2e0=etan2x×ln2buttanx0so8=etan2x×ln2tan2x×ln2=ln8tan2x×ln2=3ln2tanx=3sinx+cosxsinxcosx=tanx+1tanx13+1313+23+131=4+232=2+3answer

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