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

1 Answer
Feb 2, 2015

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.