Trigonometry — CDS Elementary Mathematics PYQs

If $\frac{\cos \theta }{1-\sin \theta }+\frac{\cos \theta }{1+\sin \theta }=4$, then which one of the following is a value of $({\tan }^2\theta +{\cot }^2\theta )$?

For $0<\theta <\frac{\pi }{2}$, consider the following:
I. $({\tan }^4\theta +{\tan }^6\theta )({\cot }^4\theta +{\cot }^6\theta )={\sec }^2\theta {\csc }^2\theta$
II. $\frac{\tan \theta +\sin \theta }{\tan \theta -\sin \theta }={\cot }^2\theta (\sec \theta +1{)}^2$

Which of the above is/are identities?

If $3\sin \theta +4\cos \theta =5$, then what is a value of $4\tan \theta +3\cot \theta$?

Consider the following for the next two (02) items:
Let $12(\tan \theta +\cot \theta )=25$, where $45^{\circ }<\theta <90^{\circ }$.

What is a value of $(\sin \theta -\cos \theta )$?

Consider the following for the next two (02) items:
Let $12(\tan \theta +\cot \theta )=25$, where $45^{\circ }<\theta <90^{\circ }$.

What is a value of $(\csc \theta +\sec \theta )$?

Consider the following for the next two (02) items:
$\csc \theta -\sin \theta =p^3$ and $\sec \theta -\cos \theta =q^3$

What is $\tan \theta$ equal to?

Consider the following for the next two (02) items:
$\csc \theta -\sin \theta =p^3$ and $\sec \theta -\cos \theta =q^3$

What is $p^4{q}^2+p^2{q}^4$ equal to?

Consider the following for the next two (02) items:
$p{\sin }^2\alpha +q{\cos }^2\alpha =m$ and $p{\cos }^2\beta +q{\sin }^2\beta =n$

What is ${\tan }^2\alpha$ equal to?

Consider the following for the next two (02) items:
$p{\sin }^2\alpha +q{\cos }^2\alpha =m$ and $p{\cos }^2\beta +q{\sin }^2\beta =n$

What is ${\cot }^2\beta$ equal to?

Consider the following for the next two (02) items:
$p+q\cot \theta =3\csc \theta$ and $q-p\cot \theta =2\csc \theta$

What is $p^2+q^2$ equal to?