Proving $(1+\sec x+\tan x)(1+\csc x+\cot x) = 2(1+\tan x+\cot x+\sec x+\csc x)$

Question

I'm having some trouble proving this trig identity, I want to prove that:

$$(1+\sec x+\tan x)(1+\csc x+\cot x) = 2(1+\tan x+\cot x+\sec x+\csc x)$$

Sadly I am stuck.

Answer 1

We need to prove that $$\left(1+\frac{1}{\cos{x}}+\frac{\sin{x}}{\cos{x}}\right)\left(1+\frac{1}{\sin{x}}+\frac{\cos{x}}{\sin{x}}\right)=2\left(1+\frac{1}{\sin{x}\cos{x}}+\frac{1}{\sin{x}}+\frac{1}{\cos{x}}\right)$$ or $$(\sin{x}+\cos{x}+1)^2=2(\sin{x}\cos{x}+\sin{x}+\cos{x}+1)$$ or $$2+2\sin{x}\cos{x}+2\sin{x}+2\cos{x}=2(\sin{x}\cos{x}+\sin{x}+\cos{x}+1),$$ which is obvious.

Answer 2

expanding the left-hand side we get $$1+\csc \left( x \right) +\cot \left( x \right) +\sec \left( x \right) +\sec \left( x \right) \csc \left( x \right) +\sec \left( x \right) \cot \left( x \right) +\tan \left( x \right) +\tan \left( x \right) \csc \left( x \right) +\tan \left( x \right) \cot \left( x \right) $$ can you simplify this?