Solve for x : If $( 1 + \tan x ) ( 1 - \tan x ) ( \sec^ 2 x ) + 2 ^ { \tan^ 2 x } = 0$

Asked Dec 04 '15 at 02:50

Active Dec 04 '15 at 03:10

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If $( 1 + \tan x ) ( 1 - \tan x ) ( \sec^ 2 x ) + 2 ^ { \tan^ 2 x } = 0$ , then find the number of solutions in the interval $( -\pi/2 , \pi/2 )$.

I did solve , but I get 2 solutions instead of 4 as stated in answer.

My solution :

$$( 1 + \tan x ) ( 1 - \tan x ) ( \sec^2 x ) + 2 ^ { \tan^ 2 x } = 0 \\ ( 1 - \tan^2 x ) ( 1 + \tan^2 x ) + 2 ^ { \tan^ 2 x } = 0 \\ 1 - \tan^4 x + 2 ^ { \tan^ 2 x } = 0$$

Let $\tan^2 x = z$

$\implies 2 ^ z = z^2 - 1$

By hit and trial ( putting random integers )

I got $z = 3$

so then $x = n\pi \pm \pi/3$.

In $( -\pi/2 , \pi/2 )$ , I got , $-\pi/3$, $\pi/3$ only i.e. 2 solutions .

But the answer in the book says , there are 4 .

edited Dec 04 '15 at 03:10

asked Dec 04 '15 at 02:50

Ricky

There is another solution for $z<0$ that is not an integer. – David Dec 04 '15 at 03:11
@David Whats that value of z then ? – Ricky Dec 04 '15 at 03:13
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draw the graphs of $z^2 $ and $2^z+1$ you will have two solutions one positive $z$ and one negative $z$ – Ekaveera Gouribhatla Dec 04 '15 at 03:13
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@EkaveeraKumarSharma No, one negative, two positive. – user236182 Dec 04 '15 at 03:13
ya you are right its two positive – Ekaveera Gouribhatla Dec 04 '15 at 03:20
I think you have to find the non-integer roots numerically. – David Dec 04 '15 at 03:20
@David How do I solve x^2 -1 = 2^x without a graphing calculator ? How do I solve it arithmetically ? – Ricky Dec 04 '15 at 03:21
@Ricky You can solve it numerically, e.g. using Newton's method. – user236182 Dec 04 '15 at 03:23
@user236182 We havent been taught about newton method . Its a high school math , isnt there any other way ? – Ricky Dec 04 '15 at 03:26
I don't know of any arithmetic way. – David Dec 04 '15 at 03:29
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http://math.stackexchange.com/questions/739145/how-to-solve-1-tan2x-sec2x2-tan2x-0?rq=1 – juantheron Dec 04 '15 at 03:43

Solve for x : If $( 1 + \tan x ) ( 1 - \tan x ) ( \sec^ 2 x ) + 2 ^ { \tan^ 2 x } = 0$

0 Answers0