Simplify:
$$\frac{1 - \sec x}{\cosec x \cdot \cot x}$$​

Correct option is C

Given
Expression = $$\frac{1 - \sec x}{\cosec x \cdot \cot x}$$​​

Formula Used
​$$\sec x = \frac{1}{\cos x} \\\ \\\csc x = \frac{1}{\sin x} \\\ \\\cot x = \frac{\cos x}{\sin x} \\\ \\\tan x = \frac{\sin x}{\cos x} \\\ \\$$​​

Solution
Convert all trigonometric terms into sine and cosine.
Numerator = $$1 - \sec x = 1 - \frac{1}{\cos x} = \frac{\cos x - 1}{\cos x}$$​​
Denominator = $$\csc x \cdot \cot x = \frac{1}{\sin x} \cdot \frac{\cos x}{\sin x} = \frac{\cos x}{\sin^2 x}$$​​

Expression = $$\frac{\frac{\cos x - 1}{\cos x}}{\frac{\cos x}{\sin^2 x}}$$​​

Expression = $$\frac{\cos x - 1}{\cos x} \cdot \frac{\sin^2 x}{\cos x}$$​​

Expression = $$(\cos x - 1) \cdot \frac{\sin^2 x}{\cos^2 x}$$​​

Since $$\frac{\sin^2 x}{\cos^2 x} = \tan^2 x:$$​​

Expression =$$(\cos x - 1) \tan^2 x$$​​
Final Answer
So the correct answer is (c)