Proving $(\sec^2x+\tan^2x)(\csc^2x+\cot^2x)=1+2\sec^2x\csc^2x$ and $\frac{\cos x}{1-\tan x}+\frac{\sin x}{1-\cot x} = \sin x + \cos x $ 
Prove the following identities: 
  $$(\sec^2 x + \tan^2x)(\csc^2 x + \cot^2x) = 1+ 2 \sec^2x \csc^2 x 
\tag i$$
$$\frac{\cos x}{1-\tan x} + \frac{\sin x}{1-\cot x} = \sin x + \cos x
\tag {ii}$$

For $(\mathrm i)$, I initially tried simplifying what was in the 2 brackets but ended up getting 1 + 1. 
I then tried just multiplying out the brackets and got as far as $$1+ \sec^2x + \frac{2}{\cos^2x \sin^2x}$$

$$(\sec^2x+\tan^2x)(\csc^2x+\cot^2x)$$
$$=(2\sec^2x-1)(2\csc^2x-1)$$
$$=4\sec^2x\csc^2x-2(\sec^2x+\csc^2x)+1$$
Now use $\sec^2x+\csc^2x=\cdots=\sec^2x\csc^2x$
The second one has been solved by Taussig
