Solve the equatios system:
$\displaystyle \left\{ \begin{array}{l}
7\cos \alpha + 9\cos \beta = 10 \\
7\sin \alpha - 9\sin \beta = 0 \\
\end{array} \right.
$
Note that you have
$\displaystyle \left\{ \begin{array}{l}
7 \sin \alpha \cos \alpha + 9 \sin \alpha \cos \beta = 10 \sin \alpha \\
7\sin \alpha \cos \alpha - 9\sin \beta \cos \alpha = 0 \\
\end{array} \right.
$
(Do you see what I did?) Now subtract the second equation from the first. can you finish from there?
Add two equations to the system:
$\displaystyle \left\{\begin{array}{l}7\cos\alpha+9\cos\beta=10\\ 7\sin\alpha-9\sin\beta=0\\\sin^2\alpha+\cos^2\alpha=1\\\sin^2\ beta+\cos^2\beta=1\end{array}\right.$
$\displaystyle \sin\beta=\frac{7\sin\alpha}{9}, \ \cos\beta=\frac{10-7\cos\alpha}{9}$
Replace them in the last equation.
$\displaystyle \frac{49\sin^2\alpha}{81}+\frac{100-140\cos\alpha+49\cos^2\alpha}{81}=1\Rightarrow \cos\alpha=\frac{17}{35}\Rightarrow\cos\beta=\frac {77}{105}$
Now find $\displaystyle \sin\alpha, \ \sin\beta$ from the last two equations and choose the sign such as the second equation is satisfied. Then find $\displaystyle \alpha, \ \beta$.