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Find a basis for the space of $2\times 2$ diagonal matrices.
$\text{Basis }=\{\left[\begin{array}{cc}& \\ & \end{array}\right],\left[\begin{array}{cc}& \\ & \end{array}\right]\}$

Let B be a 4x4 matrix to which we apply the following operations:
1. double column 1,
2. halve row 3,
3. add row 3 to row 1,
4. interchange columns 1 and 4,
5. subtract row 2 from each of the other rows,
6. replace column 4 by column 3,
7. delete column 1 (column dimension is reduced by 1).
(a) Write the result as a product of eight matrices.
(b) Write it again as a product of ABC (same B) of three matrices.

Help me to prove that
${\displaystyle \arcsin \left(\sqrt{2}\sin \left(t\right)\right)+\arcsin \left(\sqrt{\cos \left(2t\right)}\right)=\frac{\pi }{2}}$

Find $\cos 8x$ if:
$\tan x\tan 2x=\cot 2x\cot 3x$

Inverse Trigonometry System of Equations
${\displaystyle 2{\tan }^{-1}\left(\sqrt{x-x^2}\right)={\tan }^{-1}\left(x\right)+{\tan }^{-1}(1-x)}$
I have a feeling solution includes drawing triangles but cannot make the leap to get the solution

If ${\displaystyle 2(\sin a+\cos a)\sin b=3-\cos b}$, then find ${\displaystyle 3{\tan }^{2}a+4{\tan }^{2}b}$
I doubt there is a solution on the internet now, but they do have a number 35 as the final answer.
My tiny little progress:
Use simple simplification,
${\displaystyle 3{\tan }^{2}a+4{\tan }^{2}b=\frac{3}{{\cos }^{2}a}+\frac{4}{{\cos }^{2}b}-7}$
Intuitively, in order to get this, the given equation should be squared. But that is where I got stuck, because the term cosb seems hard to handle; it just seems different from other terms.

Let's say that I have two ordered sets of numbers $\{1,2\}$ and $\{3,4\}$. I'm trying to figure out the number of possible ways to combine these two sets into one without breaking the ordering of the two sets.
So for instance, $\{1,2,3,4\}$, $\{3,4,1,2\}$, and $\{1,3,2,4\}$ are valid combinations, but $\{2,1,4,3\}$ isn't. How do I figure out the number of valid combinations? This feels like something I should remember from college, but I'm drawing a blank. It feels somewhere in between a combination and a permutation.