What is the value of $\frac{\cot 224°-\cot 134°}{\cot 226°+\cot 316°}?$

Consider the following statements

$\mathrm{I}.$ $\sin \left|x\right|+\cos \left|x\right|$ is always positive.

$\mathrm{II}.$ $\sin \left(x^2\right)+\cos \left(x^2\right)$ is always positive. 

Which of the above statement(s) is/are correct?

Consider the following statements

$\mathrm{I}.$ Value of $\sin \theta$ oscillates between $-1$ and $1.$

$\mathrm{II}.$ Value of $\cos \theta$ oscillates between $0$ and $1.$

Which of the above statement(s) is/are correct?

Consider the following statements

$\mathrm{I}.$ $n\left\{{\sin }^2{\left(67\frac{1}{2}\right)}^{°}-{\sin }^2{\left(22\frac{1}{2}\right)}^{°}\right\}>1$ for all positive integers $n\ge 2$.

$\mathrm{II}.$ If $x$ is any positive real number, then $nx>1$ for all positive integers $n\ge 2$.

Which of the above statement(s) is/are correct?

Consider the following statements

$\mathrm{I}.$ If $3\theta$ is an acute angle such that $\sin 3\theta =\cos 2\theta ,$ then the measurement of $\theta$ in radian equals to $\frac{\mathrm{\pi }}{10}$

$\mathrm{II}.$ One radian is the angle subtended at the centre of a circle by an arc of the same circle whose length is equal to the diameter of that circle.

Which of the above statement(s) is/are correct?