If  $\frac{\cos \theta }{a}=\frac{\sin \theta }{b}$, then  $\frac{a}{\sec 2\theta }+\frac{b}{\cos ec2\theta }=$

$\frac{\cos \theta }{a}=\frac{\sin \theta }{b}=k$

$\cos \theta =ak,\sin \theta =bk$

$\frac{a}{\sec 2\theta }+\frac{b}{\cos ec2\theta }=a\cos 2\theta +b\sin 2\theta$

$a\cos 2\theta +b\sin 2\theta =a(1-2{\sin }^2\theta )+b2\sin \theta \cdot \cos \theta$

$=a-2a{\left(bk\right)}^2+2b\left(bk\right)\left(ak\right)$

$=a-2a{b}^2{k}^2+2b^2{k}^{2}a$

$=a$