The objective is to simplify the expression: \[ \frac{\sin(\pi + x) \cos\left(\frac{\pi}{2} + x\right) \tan\left(\frac{3\pi}{2} - x\right) \cot(2\pi - x)}{\sin(2\pi - x) \cos(2\pi + x) \csc(-x) \sin\left(\frac{3\pi}{2} + x\right)}. \]
Step 1: Simplify Individual Trigonometric Functions
Applying standard trigonometric identities yields:
\( \sin(\pi + x) = -\sin(x) \)
\( \cos\left(\frac{\pi}{2} + x\right) = -\sin(x) \)
\( \tan\left(\frac{3\pi}{2} - x\right) = \cot(x) \)
\( \cot(2\pi - x) = -\cot(x) \)
\( \sin(2\pi - x) = -\sin(x) \)
\( \cos(2\pi + x) = \cos(x) \)
\( \csc(-x) = -\csc(x) \)
\( \sin\left(\frac{3\pi}{2} + x\right) = -\cos(x) \)
Substitute these simplified terms back into the original expression.
Step 2: Substitute Simplified Terms into the Expression
The numerator simplifies to: \[ (-\sin x)(-\sin x)(\cot x)(-\cot x) = -\sin^2(x) \cot^2(x). \]
The denominator simplifies to: \[ (-\sin x)(\cos x)(-\csc x)(-\cos x) = -\sin(x) \cos^2(x) \csc(x). \]
Step 3: Combine and Simplify the Expression
The expression now is: \[ \frac{-\sin^2(x) \cot^2(x)}{-\sin(x) \cos^2(x) \csc(x)}. \]
Utilizing the identities \( \cot(x) = \frac{\cos(x)}{\sin(x)} \) and \( \csc(x) = \frac{1}{\sin(x)} \), we get: \[ \frac{\sin^2(x) \left(\frac{\cos^2(x)}{\sin^2(x)}\right)}{\sin(x) \cos^2(x) \left(\frac{1}{\sin(x)}\right)}. \]
Further simplification leads to: \[ \frac{\cos^2(x)}{\cos^2(x)} = 1. \]
Final Answer: The simplified value of the expression is: \[ \boxed{1 \, \text{(Option C)}}. \]
If
\( p \): It is raining today,
\( q \): I go to school,
\( r \): I shall meet my friends,
and \( s \): I shall go for a movie, then which of the following represents:
"If it does not rain or if I do not go to school, then I shall meet my friend and go for a movie?"