Express $7{\sin }^{2}x+4{\cos }^{2}x$ in form of $a+b{\sin }^{2}x$

State the range ol the function $f\left(x\right)=7{\sin }^{2}x+4{\cos }^{2}x$, for the domain $0⩽x⩽2\pi$.

Express $4\sin \theta -{\cos }^2\theta$ in the form $(\sin \theta +a{)}^2+b$

State the maximum and minimum values of $4\sin \theta -{\cos }^2\theta$. for the domain $0⩽\theta ⩽2\pi$.

Given that $a=\frac{1-\sin \theta }{2\cos \theta }$, show that $\frac{1}{a}=\frac{2(1+\sin \theta )}{\cos \theta }$.

Given that $a=\frac{1-\sin \theta }{2\cos \theta }$. Hence, find $\sin \theta$ and $\cos \theta$ in terms of $a$.