If $sin^{-1}(x-2)+cos^{-1}(x)+tan^{-1}(x+2)+cot^{-1}(x+4) = sec^{-1}(\sqrt{k})-\frac{π}{2}$, then $cos(2\,\text{cosec}^{-1}\sqrt{k-1}) = \ldots$

Question

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

Answer 1

Step 1: Truth table idea:
$P \rightarrow Q$ is false only when P is true and Q is false.

Step 2: So the negation says:
P is true and Q is false, which is the sentence in option (C).

Final Answer:
Option (C). \[ \boxed{\text{(C)}} \]

If \(sin^{-1}(x-2)+cos^{-1}(x)+tan^{-1}(x+2)+cot^{-1}(x+4) = sec^{-1}(\sqrt{k})-\frac{π}{2}\), then \(cos(2\,\text{cosec}^{-1}\sqrt{k-1}) = \ldots\)

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The Correct Option is B

Solution and Explanation

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