Mimi M.

asked • 08/29/20

pathogerogean identity

1+tan2(x). Explain why

this is not the same function as g(x) = cos2(x).


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Tom K. answered • 08/29/20

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This response is edited reflected the modified problem. While cos(x) exists for all x, sec(x) does not exist when cos(x) = 0. Thus, 1/1+tan^2(x) does not equal cos^2(x) when cos(x) = 0.

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Mimi M.

1/(1+tan2(x)). Explain why this is not the same function as g(x) = cos2(x).
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08/29/20

Alex E.

It’s a domain issue. The domain of tan(x) excludes any x = n*pi + pi/2 where n is an integer (....-3,-2,-1,0,1,2,3...) . This is bc tan(x) = sin(x)/ cos(x) and as we know in math division by zero is undefined. So anytime cos(x) = 0, tan(x) becomes undefined. Recall cos(x) = 0 whenever x= n*pi + pi/2. Thus any number x= n*pi + pi/2 is not in the domain of the function f(x) = 1/(1+tan^2(x)). However cos(x) is defined for every real numbers and thus g(x) = cos(x) * cos(x) is defined on the domain of all real numbers. So in conclusion g(x)’s domain is the reals while the domain of f(x) is the reals Take Away any real number x = n*pi + pi/2. Hence the domain of f(x) and g(x) given are not the same... However, I believe they would be the same if the problem restricted the domain of cos(x) to be the reals Take Away any real number x=n*pi + pi/2.
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08/30/20

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