Question

Assertion :${\csc }^{-1}(\frac{1}{2}+\frac{1}{\sqrt{2}})>{\sec }^{-1}(\frac{1}{2}+\frac{1}{\sqrt{2}})$ Reason: ${\csc }^{-1}x>{\sec }^{-1}x$ if $1\le x<\sqrt{2}$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Assertion ${\cos }^{-1}(\frac{1}{2}+\frac{1}{\sqrt{2}})>{\sec }^{-1}(\frac{1}{2}+\frac{1}{\sqrt{2}}){\sin }^{-1}\left(\frac{2\sqrt{2}}{2+\sqrt{2}}\right)>{\cos }^{-1}\left(\frac{2\sqrt{2}}{2+\sqrt{2}}\right){\sin }^{-1}\left(0.828\right)>{\cos }^{-1}\left(0.828\right)$

we know

${\sin }^{-1}x>{\cos }^{-1}x$ ($\frac{1}{\sqrt{2}}<\times \le 1$)

hence Assertion is correct

also $co{\sec }^{-1}x>{\sec }^{-1}x$ if $1\le x<\sqrt{2}$

for this value of $1\le x<\sqrt{2},\sin x<\cos x$

so $\cos x>\sec x$

reason is also correct and also correct explanation of Assertion