Question

Assertion :Equation $\sqrt{1-\sin 2x}=\sin x$ has 1 solution for $xϵ[0,\pi /4]$ Reason: $\cos x>\sin x$ when $x\in [0,\frac{\pi }{4}]$

• 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

$\sqrt{1-\sin 2x}=\sin x\Rightarrow \sqrt{{(\cos x-\sin x)}^2}=\sin x$

$\Rightarrow |\cos x-\sin x|=\sin x$

For $x\in [0,\frac{\pi }{4}]\cos x>\sin x$

$\cos x-\sin x=\sin x\Rightarrow 2\sin x=\cos x$

$\Rightarrow \tan x=\frac{1}{2}\Rightarrow x={\tan }^{-1}\frac{1}{2}$

Has only one solution in $[0,\frac{\pi }{4}]$