# Number of Solutions for Trigonometric Equation on $\theta \in [-\pi , \pi]$

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

How many solutions does the equation:

$$(1+\sec\theta)(1+\csc\theta)=0$$

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

My trial so far:

$$2\cos^{2}\left(\frac{\theta}{2}\right)=1+\cos\theta$$

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

$$1+\sec\theta=2\frac{\cos^{2}\left(\frac{\theta}{2}\right)}{\cos\theta}$$

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

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 $\cos x = -1 \implies \sin x = 0$.