<s>[INST] Odd or even function. Determine whether the following function is odd or even :
$$f(x) =(\cos{x} + \sin{x} +1)(\cos{x} + \sin{x} -1)$$
My turn :
$$f(x) = (\cos{x}+ \sin{x})^2 -1= \sin{2x}$$
Then f is an odd function
Another turn :
$$f(-x) = (\cos{x}- \sin{x} +1)(\cos{x} -\sin{x} -1)$$
But the last form seems neither odd nor even. My question is why i could not reach the same another using the last form ?
 [/INST] While it seems that everything in the original post is correct, you want to show that: 
$f(x) = -f(-x)$ 
$(\cos x + \sin  x + 1)(\cos x +\sin x - 1) = -(\cos (-x) +\sin (-x) + 1)(\cos (-x) + \sin (-x) -1)\\
(\cos x + \sin  x + 1)(\cos x +\sin x - 1) = -(\cos x -\sin x + 1)(\cos x - \sin x -1)\\
(\cos x + \sin x)^2 - 1  = -(\cos x - \sin x)^2 + 1\\
2\sin x\cos x = 2\sin x\cos x$
It is not obvious that 
$f(x) = (\cos x + \sin  x + 1)(\cos x +\sin x - 1)$ because neither of the factors are odd nor even.  Nonetheless, it is.
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