Question

Let \[f(x) = \sec x + \tan x,\] \[g(x) = \dfrac{{[\tan x]}}{{[1 - \sec x]}}\]
Statement 1: g is an odd function
Statement 2: f is neither an odd function nor an even function
(1) Statement 1 is true
(2) Statement 2 is true
(3) 1 and 2 both are true
(4) 1 and 2 both are false

Answer 1

Hint: We can determine the correct conclusion by checking the conditions for even and odd functions. we know that f is a real-valued function of a real variable. Then f is even if the following equation holds for all x such that \[f( - x) = f(x)\]. It is odd if the following equation holds for all x such that \[f( - x) = - f(x)\].

Complete step by step answer:
Now let us consider the given function
\[f(x) = \sec x + \tan x - - - \left( 1 \right)\]
Now multiply by -1 on both sides we g et
\[ - f(x) = - \sec x - \tan x - - - \left( 2 \right)\]
Now replace x by -x in equation 1 we get
\[f( - x) = \sec ( - x) + \tan ( - x)\]
We get
\[f( - x) = \sec x - \tan x - - - \left( 3 \right)\]
We note that from equations 2 and 3 we can note that \[f( - x) \ne - f(x)\] and from equation 1 and 3 we note that \[f( - x) \ne f(x)\]
Therefore, f is neither odd nor even.
Now let us consider the second equation
\[g(x) = \dfrac{{[\tan x]}}{{[1 - \sec x]}}\]
Now we can note that \[g( - x) \ne - g(x)\] as \[\sec ( - x) = \sec (x)\] therefore
The function g is odd function
Therefore, both the statements are true.

So, the correct answer is “Option 3”.

Note: In mathematics, even functions and odd functions are functions which satisfy the particular symmetry relations, with respect to taking additive inverses. Evenness and oddness are generally considered for real functions, that is real-valued functions of a real variable. However, the concepts may be more generally defined for functions whose domain and codomain both have a notion of additive inverse.
Geometrically, the graph of an even function is symmetric with respect to the y-axis, meaning that its graph remains unchanged after reflection about the y-axis.
Geometrically, the graph of an odd function has rotational symmetry with respect to the origin, meaning that its graph remains unchanged after rotation of 180 degrees about the origin.