Question

Which among the following pair(s) of function are identical.

• A. $f\left(x\right)=\frac{\sec x}{\cos x}-\frac{\tan x}{\cot x},g\left(x\right)=\frac{\cos x}{\sec x}+\frac{\sin x}{cosecx}$
• B. $f\left(x\right)=sgn(x^2-6x+10),g\left(x\right)=sgn({\cos }^{2}x+{\sin }^2(x+\frac{\pi }{3}))$ where sgn denotes signum function.
• C. $f\left(x\right)=e^{ln(x^2-5x+6)},g\left(x\right)=x^2-5x+6$
• D. $f\left(x\right)=\frac{\sin x}{\sec x}+\frac{\cos x}{cosecx},g\left(x\right)=\frac{2{\cos }^{2}x}{\cot x}$

Answer 1

Answer: (A), (B)

$f\left(x\right)=sgn(x^2-6x+10),g\left(x\right)=sgn({\cos }^{2}x+{\sin }^2(x+\frac{\pi }{3}))$ where sgn denotes signum function.

Functions $f\left(x\right)$ and $g\left(x\right)$ are said to be equal or identical function's if
$\left(i\right)$ Domain of $f\left(x\right)=g\left(x\right)$ and
$\left(ii\right)$ Range of $f\left(x\right)=g\left(x\right)$

For $f\left(x\right)=\frac{\sec x}{\cos x}-\frac{\tan x}{\cot x},g\left(x\right)=\frac{\cos x}{\sec x}+\frac{\sin x}{\text{cosec}x}$
Domain of both functions is $x\in R-\{\frac{n\pi }{2},n\in Z\}$
Also, both functions simplify to $1$ i.e. Range $=\left\{1\right\}$
Hence, both functions are identical.

For $f\left(x\right)=sgn(x^2-6x+10),g\left(x\right)=sgn({\cos }^{2}x+{\sin }^2(x+\frac{\pi }{3}))$
As $x^2-6x+10=(x-3{)}^2+1>0,{\cos }^{2}x+{\sin }^2(x+\frac{\pi }{3})>0$
Hence $f\left(x\right)=g\left(x\right)=1\forall x\in R.$ and Domain of $f\left(x\right)=g\left(x\right)=R$
Hence both functions are identical.

For $f\left(x\right)=e^{ln(x^2-5x+6)},g\left(x\right)=x^2-5x+6$
Domain of $f\left(x\right)$ is $x\in (-\infty ,2)\cup (3,\infty )$ but domain of $g\left(x\right)=R$
Hence, they are not identical functions.

For
$f\left(x\right)=\frac{\sin x}{\sec x}+\frac{\cos x}{\text{cosec}x},g\left(x\right)=\frac{2{\cos }^{2}x}{\cot x}$
$f\left(x\right)=\frac{\sin x}{\sec x}+\frac{\cos x}{\text{cosec}x}$
$=2\sin x\cos x$
$=\frac{2{\cos }^{2}x}{\cot x}$
$=g\left(x\right)$
And domain of both the functions is also same.
Hence, they are identical functions.