How do you prove that cos2x(1+tan2x)=1?

1 Answer
Feb 2, 2015

The answer2 are: x=kπ and π4+kπ.

The equation can be written:

cos2x+cos2xtan2x=1cos2x+cos2xsin2xcos2x=1cos2x+sin2x=1sin2x+cos2x=1.

Now it is possible multiply both members for 22:

22sin2x+22cos2x=22

sin2xcos(π4)+cos2xsin(π4)=22.

Using the addition formula:

sin(2x+π4)=22.

The sinus is 22 if its argument is π4+2kπ or 34π+2kπ.

So:

2x+π4=π4+2kπ2x=2kπx=kπ,

and

2x+π4=34π+2kπ2x=π2+2kπx=π4+kπ.