Trigonometry — CDS Elementary Mathematics PYQs

If $2{\sin }^4\alpha +2{\cos }^4\alpha -1=0$, where $0\le \alpha <\frac{\pi }{2}$, then what is $\sin 2\alpha +\cos 2\alpha$ equal to?

Consider the following:
I. $1-{\sin }^6\alpha ={\cos }^2\alpha ({\cos }^4\alpha -3{\cos }^2\alpha +3)$
II. ${\cos }^8\alpha -{\sin }^8\alpha =2{\sin }^2\alpha (1-{\cos }^4\alpha +{\sin }^2\alpha {\cos }^2\alpha )$
Which of the above is/are identities?

If $p=\frac{1}{\csc \theta +\cot \theta }$ and $q=\csc \theta$, then what is $p^2-2pq$ equal to?

Consider the following statements:
I. $(\csc \alpha -\sec \alpha )$ is always positive in the first quadrant.
II. $(\tan \alpha -\cot \alpha )$ is always negative in the first quadrant.
Which of the statements given above is/are correct?

What is $\frac{\sin \theta }{1-\cot \theta }+\frac{\cos \theta }{1-\tan \theta }$ $(\theta \ne \frac{\pi }{4})$ equal to?

If ${\cot }^2\theta -3\sqrt{3}\cot \theta +6=0$, where $\frac{\pi }{6}\le \theta <\frac{\pi }{2}$, then what is a value of $\sin \theta +\cos 2\theta$?

Which of the following equations is/are possible?
I. ${\sin }^2\theta =\frac{(x+y{)}^2}{4xy}$, where $x$, $y$ are positive unequal real quantities.
II. $\sin \theta +\cos \theta =x+\frac{1}{x}$, where $x$ is a positive real quantity.
Select the correct answer using the code given below:

If $m^2(\sin \theta -1)+n^2(\sin \theta +1)=0$, where $0<\theta <\frac{\pi }{2}$, then what is $(m^2+n^2)\cos \theta -(m^2-n^2)\cot \theta$ equal to?

If $\sin \alpha +\cos \alpha =\sqrt{2}$, where $0<\alpha <\frac{\pi }{2}$, then what is ${\sin }^3\alpha -{\cos }^3\alpha$ equal to?

What is $(1+\cot \alpha -\csc \alpha )(1+\tan \alpha +\sec \alpha )$ equal to?