Question

the final simplified expression should be.\
\\(\frac{1}{\sin x \cos x} - \cot x\\)\
\\(= \frac{1}{\sin x \cos x} - \frac{\sin x}{\cos x}\\)\
\\(= \frac{1}{\sin x \cos x} - \frac{\sin^2 x}{\sin x \cos x}\\)\
\\(= \frac{1 - \sin^2 x}{\sin x \cos x}\\)\
\\(= \frac{\cos^2 x}{\sin x \cos x}\\)\
\\(= \frac{\cos x}{\sin x}\\)\
\\(= \cot x\\)\
\
\\(\bigcirc\\) the mistake is \\(\cot x\\) was written as \\(\frac{\sin x}{\cos x}\\) instead of \\(\frac{\cos x}{\sin x}\\). the final answer is \\(\tan x\\).\
\\(\bigcirc\\) the mistake is going from here \\(\frac{1 - \sin^2 x}{\sin x \cos x}\\) to here \\(\frac{\cos^2 x}{\sin x \cos x}\\).the final answer is \\(\sin x\\).\
\\(\bigcirc\\) the mistake is \\(\cot x\\) was written as \\(\frac{\sin x}{\cos x}\\) instead of \\(\frac{\cos x}{\sin x}\\). the final answer is \\(\sin x\\).\
\\(\bigcirc\\) the mistake is going from here \\(\frac{1 - \sin^2 x}{\sin x \cos x}\\) to here \\(\frac{\cos^2 x}{\sin x \cos x}\\).the final answer is \\(\tan x\\).

Explanation:

Step1: Recall cotangent definition

$\cot x = \frac{\cos x}{\sin x}$

Step2: Identify the error

In the given work, $\cot x$ was incorrectly rewritten as $\frac{\sin x}{\cos x}$ (this is $\tan x$, not $\cot x$).

Step3: Correct the simplification

Start with the original expression:
$\frac{1}{\sin x \cos x} - \cot x = \frac{1}{\sin x \cos x} - \frac{\cos x}{\sin x}$
Rewrite to get a common denominator:
$= \frac{1}{\sin x \cos x} - \frac{\cos^2 x}{\sin x \cos x}$
Combine the fractions:
$= \frac{1 - \cos^2 x}{\sin x \cos x}$
Use Pythagorean identity $1-\cos^2 x = \sin^2 x$:
$= \frac{\sin^2 x}{\sin x \cos x}$
Simplify the fraction:
$= \frac{\sin x}{\cos x} = \tan x$

Answer:

The mistake is $\cot x$ was written as $\frac{\sin x}{\cos x}$ instead of $\frac{\cos x}{\sin x}$. The final answer is $\tan x$.