Question #18682

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Answer 1 · 2017-06-26T07:20:51

$tan(x/2) *(1+secx)*(1+sec2x)*(1+sec2^2x)*...(1+sec2^nx)$

$=tan(x/2)*(1+cosx)/cosx*(1+sec2x)*(1+sec2^2x)*...(1+sec2^nx)$

$=sin(x/2)/cos(x/2)*(2cos^2(x/2))/cosx*(1+sec2x)*(1+sec 2^2x)*...(1+sec2^nx)$

$=(2sin(x/2)cos(x/2))/cosx*(1+sec2x)*(1+sec 2^2x)*...(1+sec2^nx)$

$=sinx/cosx*(1+cos2x)/cos(2x)*(1+sec 2^2x)*...(1+sec2^nx)$

$=sinx/cosx*(2cos^2x)/cos(2x)*(1+sec 2^2x)*...(1+sec2^n x)$

$=(2sinxcosx)/cos(2x)*(1+sec 2^2x)*...(1+sec 2^nx)$

$=(sin2x)/cos(2x)*(1+sec 2^2 x)*...(1+sec2^nx)$

$=tan2x*(1+sec 2^2x)*...(1+sec2^nx)$

$=.............$

$=tan(2^n x)$