functions and domain

Question

1/(1+tan2(x)). Explain why

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

Answer 1

cos²x is defined when x = π/2 whereas 1 / (1 + tan²x) is not defined for that x value.

Note: 1 + tan²x = sec²x So, 1 / (1 + tan²x) = 1 / sec²x = cos²x.

1 / (1 + tan²x) and cos²x are equivalent for all x except when 1 / (1 + tan²x) is undefined.

Answer 2

 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(x))^2). However cos(x) is defined for every real number 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 g(x) to be the reals Take Away any real number x=n*pi + pi/2.