I just ran into this question for an admission test...

How many solutions does the equation:


have for $\theta \in [-\pi , \pi]$?

My trial so far:


Divide both sides by $\cos\theta$ and I get:


Substituting and doing some algebra, I get that the solution would be $\sin\theta=-1$, which gives me $\theta=-\frac{\pi}{2}$.

However the correct solution is that there are no solutions in the interval...

What am I missing here? Any help is appreciated

Thanks in advance


The points where $\sec \theta = -1$ are exactly the points where $\csc \theta$ is undefined, and likewise for the other term. This is a consequence of the identity $\sin^2 x + \cos^2 x = 1$, so that $\cos x = -1 \implies \sin x = 0$.

  • $\begingroup$ Got it! So the correct procedure would be to check in the original equation if the possible root(s) won't cause any problems? $\endgroup$ – bertozzijr Sep 5 '17 at 19:14
  • 1
    $\begingroup$ Yes, that's right. $\endgroup$ – user296602 Sep 5 '17 at 19:15

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