Question

value from tan x-1/cos 2x

Answer 1

cos(2x)=cos²(x)-sin²(x)=(cos(x)-sin(x))(cos(x)+sin(x)).

tan(x)-1=sin(x)/cos(x)-1=(sin(x)-cos(x))/cos(x).

Therefore:

(tan(x)-1)/cos(2x)=(sin(x)-cos(x))/(cos(x)(cos(x)-sin(x))(cos(x)+sin(x)))=

-1/(cos(x)(cos(x)+sin(x)))=-1/(cos²(x)+sin(x)cos(x)).

Since cos(2x)=2cos²(x)-1, cos²(x)=(1+cos(2x))/2; and sin(2x)=2sin(x)cos(x), this expression can be written:

-2/(sin(2x)+cos(2x)+1).