Value of $\sec^2 a+2\sec^2 b$

Question

If $a,b$ are $2$ real number such that $2\sin a \sin b +3\cos b+6\cos a\sin b=7,$ Then $\sec^2 a+2\sec^2 b$ is

Try: i am trying to sve it using cauchy schwarz inequality

$$\bigg[(\sin b)(2\sin a+6\cos a)+(\cos b)(3)\bigg]^2\leq (\sin^2b+\cos^2b)\bigg((2\sin a+6\cos a)^2+3^2\bigg)$$

Could some help me to solve it , thanks

Answer 1

Hint:

$2\sin a+6\cos a=\sqrt{40}\sin (a+x)$ where $x=\sin^{-1}\frac{6}{\sqrt{40}}$.

\begin{align*} 2\sin a \sin b +3\cos b+6\cos a\sin b&=\sqrt{40}\sin (a+x)\sin b+3\cos b\\ &=\sqrt{40\sin^2(a+x)+9}\cdot \sin(b+y) \end{align*}

where $\sqrt{40}\sin(a+x)=\sqrt{40\sin^2(a+x)+9}\cdot\cos y$ and $3=\sqrt{40\sin^2(a+x)+9}\cdot\sin y$.

Since the maximum value of $\sqrt{40\sin^2(a+x)+9}\cdot\sin(b+y)$ is $7$, we need $\sqrt{40\sin^2(a+x)+9}=7$ and $\sin (b+y)=1$.

The answer should be $12$.