Question

If \( x \neq 0 \), then \[ \frac{\sin(\pi + x)\cos\left(\frac{\pi}{2} + x\right)\tan\left(\frac{3\pi}{2} - x\right)\cot(2\pi - x)}{\sin(2\pi - x)\cos(2\pi + x)\csc(-x)\sin\left(\frac{3\pi}{2} + x\right)} = \]

• A. \( 0 \)
• B. \( -1 \)
• C. \( 1 \)
• D. \( 2 \)

Hint:

To simplify trigonometric expressions, apply standard identities such as \( \sin(\pi + x) = -\sin x \), and ensure careful cancellation of like terms.

Answer 1

The Correct Option is C

We are tasked with simplifying the following 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 Trigonometric Terms

Using well-known trigonometric identities, we have:

\( \sin(\pi + x) = -\sin(x), \)

\( \cos\left(\frac{\pi}{2} + x\right) = -\sin(x), \)

\( \tan\left(\frac{3\pi}{2} - x\right) = -\cot(x), \)

\( \cot(2\pi - x) = -\cot(x), \)

\( \sin(2\pi - x) = -\sin(x), \)

\( \cos(2\pi + x) = \cos(x), \)

\( \csc(-x) = -\csc(x), \)

\( \sin\left(\frac{3\pi}{2} + x\right) = -\cos(x). \)

Now, substitute these identities into the original 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 2: Substitute the Simplified Terms

The numerator becomes: \[ \sin(\pi + x) \cdot \cos\left(\frac{\pi}{2} + x\right) \cdot \tan\left(\frac{3\pi}{2} - x\right) \cdot \cot(2\pi - x) = (-\sin x)(-\sin x)(-\cot x)(-\cot x). \]

Simplifying this gives: \[ \text{Numerator} = \sin^2(x) \cdot \cot^2(x). \]

The denominator simplifies as follows: \[ \sin(2\pi - x) \cdot \cos(2\pi + x) \cdot \csc(-x) \cdot \sin\left(\frac{3\pi}{2} + x\right) = (-\sin x)(\cos x)(-\csc x)(-\cos x). \]

Simplifying this gives: \[ \text{Denominator} = \sin(x) \cdot \cos^2(x) \cdot \csc(x). \]

Step 3: Simplify the Final Expression

Now, the expression becomes: \[ \frac{\sin^2(x) \cdot \cot^2(x)}{\sin(x) \cdot \cos^2(x) \cdot \csc(x)}. \]

Substitute \( \cot(x) = \frac{\cos(x)}{\sin(x)} \) and \( \csc(x) = \frac{1}{\sin(x)} \) into the equation: \[ \frac{\sin^2(x) \cdot \left(\frac{\cos^2(x)}{\sin^2(x)}\right)}{\sin(x) \cdot \cos^2(x) \cdot \frac{1}{\sin(x)}}. \]

Simplify the expression: \[ \frac{\cos^2(x)}{\cos^2(x)} = 1. \]

Final Answer: The value of the expression simplifies to: \[ \boxed{1 \, \text{(Option C)}}. \]