Question

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

• A. \(\frac{15}{16}\)
• B. \(\frac{31}{32}\)
• C. \(\frac{63}{64}\)
• D. \(\frac{7}{8}\)

Hint:

The negation of if P then Q is P and not Q.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The statement is "if P then Q" where P is "an integer is greater than 4 and less than 5" and Q is "it is a multiple of 3". The negation of \(P \rightarrow Q\) is \(P \wedge \sim Q\).

Step 2: Apply:
Keep P as it is and negate Q: "An integer is greater than 4 and less than 5 but it is not a multiple of 3".

Step 3: Check the options:
Option (A) changes P, option (B) is another conditional, and option (D) negates P as well as Q. Only (C) keeps P and negates Q.

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