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

I have to find number of solutions of this equation such that $\frac{-\pi}{2}\lt\theta\lt\frac{\pi}{2}$.

I am not getting how to approach this question. Any help would be appreciated, thanks!

I was able to reduce it to $2^{tan^2\theta} = \tan^4\theta-1 $ but I am not able to proceed further.

  • $\begingroup$ This is not only trigonometry: unless some slick trick is involved here (or, of course, unless something basic is being missed by me), this looks like a rather hard, very hard, trigonometric-exponential (a transcendental) equation. Are you sure of that $\;2^{\tan^2\theta}\;$ thing there? $\endgroup$ – Timbuc Mar 13 '15 at 9:00
  • $\begingroup$ Yes, I am sure. And it is not supposed to be extremely tough.. I'm still in high school so.. $\endgroup$ – Arpan Mar 13 '15 at 9:01
  • $\begingroup$ That must be one high school...or perhaps just a mistake of the teacher, or maybe you're supposed to do some graphing and evaluate approximately. Good luck! $\endgroup$ – Timbuc Mar 13 '15 at 9:02
  • $\begingroup$ @Timbuc It's simple trig for simplification, after that it looks like only approximation can be done... no way can a human manually compute the solutions for this answer with good precision.. I checked on WA myself.. $\endgroup$ – Kugelblitz Mar 13 '15 at 9:04
  • $\begingroup$ @Kugelblitz Yes, the first trigonometric simplifications are pretty simple. The trigo-exponential outcome is awful, and it looks to me rather odd to put a high school student to deal with such ugly things, unless he was told to do some graphing and etc. (and even it isn't an easy graph if done without some program) $\endgroup$ – Timbuc Mar 13 '15 at 9:10

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

(One more hint: Let $\tan^2\theta = x$ Then equation is: $$1-x^2+2^x=0$$ Solve for x, then solve for $\theta ).$

Edit: I checked the results on WA; Doesn't look like it's easily solvable by hand... So yes, graphing or a computational engine might be necessary... (Enter your equation here... it gives four plausible answers.. http://www.wolframalpha.com/widget/widgetPopup.jsp?p=v&id=bc455327d0772719486c1a3ecf2e96d3&title=Math%20Help%20Boards%3A%20Equation%20Solver&theme=blue)

  • $\begingroup$ I could reach until this point too. But then how do i proceed? Should I try graphing? $\endgroup$ – Arpan Mar 13 '15 at 8:59
  • $\begingroup$ @ArpanBanerjee I made an edit. $\endgroup$ – Kugelblitz Mar 13 '15 at 9:02
  • $\begingroup$ Also @ArpanBanerjee were did you come across this? Which textbook? Also, are you in CBSE? $\endgroup$ – Kugelblitz Mar 13 '15 at 9:04
  • $\begingroup$ Yes, I am in CBSE. I am currently preparing for IIT-JEE and this question was in a book by the coaching institute I go to.. $\endgroup$ – Arpan Mar 13 '15 at 9:06
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    $\begingroup$ @ArpanBanerjee The answer provided by your textbook is correct . Your work was correct , then KugelBlitz's substitution was correct then Claude's inspection yielded 3 as x and hence $\theta$ has 2 values (+ and - $tan(\theta)$) . And all this is done without a graphic calculator and so it is an appropriate question . $\endgroup$ – Klosew Mar 18 '15 at 14:47

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