How do you find the domain and range of #tan(cos(x))#?

the range of #tan(cosx is [tan(-1), tan 1]= [-1.5574, 1.5574], nearly.

Of course, the function is defined for this range,

graph{arctany=cosx [-10, 10, -5, 5]}

graph{arctany-cosx=0 [-3.14, 3.14, -1.57, 1.57]}

.

To find the domain and range of ( \tan(\cos(x)) ), we need to consider the domains and ranges of the individual functions involved, which are (\cos(x)) and (\tan(x)).

The domain of (\cos(x)) is all real numbers, as cosine is defined for all values of (x).

However, the range of (\cos(x)) is limited to the interval ([-1, 1]), as cosine values oscillate between -1 and 1 for all real (x).

Now, the tangent function, (\tan(x)), has a domain where the cosine function is not equal to zero, because the tangent function is undefined when cosine is zero (since tangent is sine divided by cosine). Therefore, the domain of (\tan(\cos(x))) is where ( \cos(x) \neq 0 ).

Since the range of (\cos(x)) is ([-1, 1]), the values of (x) for which (\cos(x) = 0) are (x = \frac{\pi}{2} + n\pi) for any integer (n). Therefore, we must exclude these values from the domain of ( \tan(\cos(x)) ).

So, the domain of ( \tan(\cos(x)) ) is all real numbers except for (x = \frac{\pi}{2} + n\pi), where (n) is an integer.

For the range of ( \tan(\cos(x)) ), since tangent is unbounded, its range is all real numbers.

In summary, the domain of ( \tan(\cos(x)) ) is all real numbers except for (x = \frac{\pi}{2} + n\pi), and the range is all real numbers.