CDS Mathematics Trigonometric Functions Medium Questions

If $p=\frac{\sin \theta }{1+\cos \theta +\sin \theta }$ and $q=\frac{1+\sin \theta }{1+\sin \theta -\cos \theta }$, what is $\left(p+\frac{1}{q}\right)\left(q+\frac{1}{p}\right)$ equal to ?

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

Let $12(\tan \theta +\cot \theta )=25$, where $45^{\circ }<\theta <90^{\circ }$. What is a value of $(\sin \theta -\cos \theta )$ ?

If $p{\sin }^2\beta +q{\cos }^2\beta =m$ and $p{\cos }^2\beta +q{\sin }^2\beta =n$. What is ${\cot }^2\beta$ equal to ?

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: $\mathrm{cosec}\theta -\sin \theta =p^3$ and $\sec \theta -\cos \theta =q^3$. What is $\tan \theta$ equal to ?

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

If $p=\frac{\sin \theta }{1+\cos \theta +\sin \theta }$ and $q=\frac{1+\sin \theta }{1+\sin \theta -\cos \theta }$. Which one of the following is correct ?

If $p+q\cot \theta =3\mathrm{cosec}\theta$ and $q-p\cot \theta =2\mathrm{cosec}\theta$. What is $p^2+q^2$ equal to ?

Let $p{\sin }^2\alpha +q{\cos }^2\alpha =m$, $q{\sin }^2\beta +p{\cos }^2\beta =n$; $p\ne m,n$ and $q\ne m,n$.

What is ${\left(\frac{\tan \alpha }{\tan \beta }\right)}^2$ equal to?

Let $p{\sin }^2\alpha +q{\cos }^2\alpha =m$, $q{\sin }^2\beta +p{\cos }^2\beta =n$; $p\ne m,n$ and $q\ne m,n$.

If $\alpha$ and $\beta$ are complementary angles, then which one of the following is correct?

Let $\mathrm{cosec}\theta -\sin \theta =p$ and $\sec \theta -\cos \theta =q$.

What is $(p\sin \theta +q\cos \theta )$ equal to?

If $\cot \theta =\sqrt{7}$, then what is $\frac{{\mathrm{cosec}}^2\theta -{\sec }^2\theta }{{\mathrm{cosec}}^2\theta +{\sec }^2\theta }$ equal to?

If $12(\tan \theta +\cot \theta )=25$ with $45^{\circ }<\theta <90^{\circ }$, what is a value of $(\mathrm{cosec}\theta +\sec \theta )$ ?

Let $\mathrm{cosec}\theta -\sin \theta =p$ and $\sec \theta -\cos \theta =q$.

What is $p^2{q}^2(p^2+q^2+3)$ equal to?

If $\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}+\dots }}}=\mathrm{cosec}\theta$, then what is $\sin \theta$ equal to?

If $8\sin \theta -\cos \theta =4$, where $0<\theta <\pi /2$, then what is $\mathrm{cosec}\theta$ equal to?

If $2\tan \theta ={\sec }^2\theta -2$, where $0<\theta <\pi /2$, then what is $\cot \theta$ equal to?

For the next two items that follow: $p{\sin }^2\alpha +q{\cos }^2\alpha =m$ and $p{\cos }^2\alpha +q{\sin }^2\alpha =n$. What is ${\tan }^2\alpha$ equal to ?

What is $(\sec \theta -\tan \theta )-\sqrt{\frac{1-\sin \theta }{1+\sin \theta }}$ equal to?