How do you prove that $cos 2x(1 + tan 2x) = 1$ ?

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Answer 1 · 2015-02-02T11:22:54

The answer2 are: $x=kpi$ and $pi/4+kpi$ .

The equation can be written:

$cos2x+cos2xtan2x=1rArrcos2x+cos2x(sin2x)/(cos2x)=1rArrcos2x+sin2x=1rArrsin2x+cos2x=1$ .

Now it is possible multiply both members for $sqrt2/2$ :

$sqrt2/2sin2x+sqrt2/2cos2x=sqrt2/2rArr$

$sin2xcos(pi/4)+cos2xsin(pi/4)=sqrt2/2$ .

Using the addition formula:

$sin(2x+pi/4)=sqrt2/2$ .

The sinus is $sqrt2/2$ if its argument is $pi/4+2kpi$ or $3/4pi+2kpi$ .

So:

$2x+pi/4=pi/4+2kpirArr2x=2kpirArrx=kpi$ ,

and

$2x+pi/4=3/4pi+2kpirArr2x=pi/2+2kpirArrx=pi/4+kpi$ .

How do you prove that cos 2x(1 + tan 2x) = 1cos 2x(1 + tan 2x) = 1?