Question

sec^2 theta = 2 + 2 tan theta,find tan theta

Answer 1

@MrNood

Answer 2

so, an identity proof.

Answer 3

I GOT
sec^4 theta - 4sec theta - 4 sec^2 theta + 4 = 0

Answer 4

sre you sure you typed the exercise correctly?

Answer 5

The question?Yes.

Answer 6

I believe there should be something else in that one.

Answer 7

\[\sec^2 \theta = 2 +2\tan \theta\]

Answer 8

I end up getting 2 sec^2 theta

Answer 9

What mistake are we making?

Answer 10

Can I please see your work?

Answer 11

I thin I did a wrong thing.

Answer 12

so we know that sec theta = 1/cos theta

than sec^2 theta = 1 / cos^2 theta = (sin^2 theta + cos^2 theta )/cos^2 theta =
=sin^2 theta /cos^2 theta +cos^2 theta /cos^2 theta = tan^2 theta +1

continuing this way i think we can getting this needed 2+2tan theta

@ganeshie8 ?

Answer 13

well tan^2 theta +1 is already sec^2 theta

Answer 14

tan^2 theta +1 = 2+2tan theta

Nice! I think that is just a quadratic which can be solved easily !

Answer 15

Isn't it an identity proof?

Answer 16

I too thought it was an identity proof. Are you looking for general solutions?

Answer 17

@Owlcoffee it isn't
He forgot to mention,find tan theta

Answer 18

yes @ganeshie8 i think you know my knowledge better from all these OS users -

and thank you your opinion - have a nice day ! to everybody

Answer 19

ah, I misread.
Then starting from:
\[\tan^2 \theta +1 =2+2\tan \theta \iff \tan^2 \theta -2\tan \theta -1=0\]
calling tan theta = g
\[g^2-2g-1=0 \]
\[g_1=1-\sqrt 2 \]
\[g_2 = 1+ \sqrt 2 \]

Answer 20

How did I get 1 and -1 then?

Answer 21

@Owlcoffee

Answer 22

just moving everything to the LHS

Answer 23

I also got

g^2 - 2g - 1 = 0

Answer 24

Yes, now you have to solve it as you would solve any quadratic equation.

Answer 25

I got 1 and -1

Answer 26

I put the solutions for it above, the results are, if you apply the quadratic formula:
\[g=\frac{ 2 \pm \sqrt{2^2-4(1)(-1)} }{ 2 }\]
\[g=\frac{ 2 \pm \sqrt{8} }{ 2 } \iff g=\frac{ 2 }{ 2 } \pm \frac{ 2\sqrt2 }{ 2 }\]
\[g=1\pm \sqrt 2\]

Answer 27

Just deconstruct the change of variables and you'll obtain the result.