Find the smallest positive values of $\theta$ and $\phi$ Question: If $\tan(\theta - \phi)=1$ and $\sec(\theta + \phi)=\frac{2}{\sqrt 3}$, Find the smallest positive values of $\theta$ and $\phi$.
My attempt: 
$\tan(\theta - \phi)=1=\tan 45^ \circ$ 
$\implies\theta-\phi=45^\circ $ (equation-1)
$\sec(\theta+\phi)=\frac{2}{\sqrt 3}$
$\implies \theta+\phi=30^\circ$ (equation-2)
Adding equation-1 and equation-2 we get, $2\theta=75^\circ$
$\implies\theta=37.5^\circ$
Substituting the value of theta in equation-1 we get, $\phi=-7.5^\circ$
The value of $\phi$ comes out to be negative but the smallest positive value of $\phi$ is required. Also the value of $\theta$ does not match with the answer. I cannot understand how to proceed further. Please help. 
 A: Note: $\theta$ and $\phi$ do not take their smallest positive values simultaneously!
$\tan(\theta-\phi)=1$ $\implies$ $\theta-\phi=(180m+45)^\circ,\ m\in\mathbb Z$.
$\sec(\theta+\phi)=\frac2{\sqrt3}$ $\implies$ $\theta+\phi=(360n\pm30)^\circ,\ n\in\mathbb Z$.
Hence, adding,
$2\theta=(180m+360n+75)^\circ$ or $(180m+360n+15)^\circ$
$\implies$ $\theta=(90m+180n+37.5)^\circ$ or $\theta=(90m+180n+7.5)^\circ$
Since $90m+180n$ can only vary in integer multiples of $90$, the smallest positive value of $\theta$ is $7.5^\circ$. (A corresponding value of $\phi$ will be $-37.5^\circ$.)
Similarly, subtracting the first equation from the second gives
$2\phi=(360n-180m-15)^\circ$ or $(360n-180m-75)^\circ$
$\implies$ $\phi=(180n-90m-7.5)^\circ$ or $\phi=(180n-90m-37.5)^\circ$
The smallest positive value of $\phi$ is clearly $90^\circ-37.5^\circ=52.5^\circ$. (This is obtained by taking e.g. $m=n=1$ in the latter possibility. A corresponding value of $\theta$ will be $277.5^\circ$.)
Just remember that $\theta$ and $\phi$ do not take their smallest positive values simultaneously.
