Answer :
To answer whether the given statements are true (T) or false (F), we need to analyze each statement individually:
### Statement 1: The domain of [tex]\(\tan(x)\)[/tex] is all real numbers.
The function [tex]\(\tan(x)\)[/tex] is defined as [tex]\(\tan(x) = \frac{\sin(x)}{\cos(x)}\)[/tex]. The domain of [tex]\(\tan(x)\)[/tex] will thus be all real numbers for which [tex]\(\cos(x) \neq 0\)[/tex]. The values of [tex]\(x\)[/tex] where [tex]\(\cos(x) = 0\)[/tex] are [tex]\(x = \frac{(2n+1)\pi}{2}\)[/tex] for any integer [tex]\(n\)[/tex]. Therefore, [tex]\(\tan(x)\)[/tex] is not defined at these points, and the domain of [tex]\(\tan(x)\)[/tex] is not all real numbers. Thus, the statement is False.
- Answer: [tex]\( F \)[/tex]
### Statement 2: [tex]\( y = \tan(2x) \)[/tex] has period [tex]\( \pi \)[/tex].
The period of the standard tangent function [tex]\( y = \tan(x) \)[/tex] is [tex]\( \pi \)[/tex], which means it repeats its values every [tex]\( \pi \)[/tex] units. However, for [tex]\( y = \tan(2x) \)[/tex], the period will be scaled by a factor of [tex]\(\frac{1}{2}\)[/tex]. This results in a period of [tex]\( \frac{\pi}{2} \)[/tex]. Hence, the statement that the period is [tex]\( \pi \)[/tex] is False.
- Answer: [tex]\( F \)[/tex]
### Statement 3: [tex]\(\tan(x)\)[/tex] and [tex]\(\cot(x)\)[/tex] are odd functions.
A function [tex]\( f(x) \)[/tex] is odd if [tex]\( f(-x) = -f(x) \)[/tex]. For [tex]\(\tan(x)\)[/tex],
[tex]\[ \tan(-x) = -\tan(x) \][/tex]
and for [tex]\(\cot(x)\)[/tex],
[tex]\[ \cot(-x) = -\cot(x) \][/tex]
These properties confirm that both [tex]\(\tan(x)\)[/tex] and [tex]\(\cot(x)\)[/tex] are odd functions. Thus, the statement is True.
- Answer: [tex]\( T \)[/tex]
### Statement 4: [tex]\(\sec(x)\)[/tex] and [tex]\(\tan(x)\)[/tex] are undefined for the same values of [tex]\(x\)[/tex].
The secant function [tex]\(\sec(x) = \frac{1}{\cos(x)}\)[/tex] is undefined where [tex]\(\cos(x) = 0\)[/tex]. Similarly, as previously discussed, [tex]\(\tan(x) = \frac{\sin(x)}{\cos(x)}\)[/tex] is also undefined where [tex]\(\cos(x) = 0\)[/tex]. These points are [tex]\(x = \frac{(2n+1)\pi}{2}\)[/tex] for any integer [tex]\(n\)[/tex]. Therefore, [tex]\(\sec(x)\)[/tex] and [tex]\(\tan(x)\)[/tex] are indeed undefined for the same values of [tex]\(x\)[/tex]. Thus, the statement is True.
- Answer: [tex]\( T \)[/tex]
Therefore, the final answers to the statements are:
1. [tex]\( F \)[/tex]
2. [tex]\( F \)[/tex]
3. [tex]\( T \)[/tex]
4. [tex]\( T \)[/tex]
### Statement 1: The domain of [tex]\(\tan(x)\)[/tex] is all real numbers.
The function [tex]\(\tan(x)\)[/tex] is defined as [tex]\(\tan(x) = \frac{\sin(x)}{\cos(x)}\)[/tex]. The domain of [tex]\(\tan(x)\)[/tex] will thus be all real numbers for which [tex]\(\cos(x) \neq 0\)[/tex]. The values of [tex]\(x\)[/tex] where [tex]\(\cos(x) = 0\)[/tex] are [tex]\(x = \frac{(2n+1)\pi}{2}\)[/tex] for any integer [tex]\(n\)[/tex]. Therefore, [tex]\(\tan(x)\)[/tex] is not defined at these points, and the domain of [tex]\(\tan(x)\)[/tex] is not all real numbers. Thus, the statement is False.
- Answer: [tex]\( F \)[/tex]
### Statement 2: [tex]\( y = \tan(2x) \)[/tex] has period [tex]\( \pi \)[/tex].
The period of the standard tangent function [tex]\( y = \tan(x) \)[/tex] is [tex]\( \pi \)[/tex], which means it repeats its values every [tex]\( \pi \)[/tex] units. However, for [tex]\( y = \tan(2x) \)[/tex], the period will be scaled by a factor of [tex]\(\frac{1}{2}\)[/tex]. This results in a period of [tex]\( \frac{\pi}{2} \)[/tex]. Hence, the statement that the period is [tex]\( \pi \)[/tex] is False.
- Answer: [tex]\( F \)[/tex]
### Statement 3: [tex]\(\tan(x)\)[/tex] and [tex]\(\cot(x)\)[/tex] are odd functions.
A function [tex]\( f(x) \)[/tex] is odd if [tex]\( f(-x) = -f(x) \)[/tex]. For [tex]\(\tan(x)\)[/tex],
[tex]\[ \tan(-x) = -\tan(x) \][/tex]
and for [tex]\(\cot(x)\)[/tex],
[tex]\[ \cot(-x) = -\cot(x) \][/tex]
These properties confirm that both [tex]\(\tan(x)\)[/tex] and [tex]\(\cot(x)\)[/tex] are odd functions. Thus, the statement is True.
- Answer: [tex]\( T \)[/tex]
### Statement 4: [tex]\(\sec(x)\)[/tex] and [tex]\(\tan(x)\)[/tex] are undefined for the same values of [tex]\(x\)[/tex].
The secant function [tex]\(\sec(x) = \frac{1}{\cos(x)}\)[/tex] is undefined where [tex]\(\cos(x) = 0\)[/tex]. Similarly, as previously discussed, [tex]\(\tan(x) = \frac{\sin(x)}{\cos(x)}\)[/tex] is also undefined where [tex]\(\cos(x) = 0\)[/tex]. These points are [tex]\(x = \frac{(2n+1)\pi}{2}\)[/tex] for any integer [tex]\(n\)[/tex]. Therefore, [tex]\(\sec(x)\)[/tex] and [tex]\(\tan(x)\)[/tex] are indeed undefined for the same values of [tex]\(x\)[/tex]. Thus, the statement is True.
- Answer: [tex]\( T \)[/tex]
Therefore, the final answers to the statements are:
1. [tex]\( F \)[/tex]
2. [tex]\( F \)[/tex]
3. [tex]\( T \)[/tex]
4. [tex]\( T \)[/tex]
