Question

Assertion :If $2\sin 2x-\cos 2x=1,x\ne (2n+1)\frac{\pi }{2}$, n is an integer, then $\sin 2x+\cos 2x$ is equal to $\frac{1}{5}$. Reason: $\sin 2x+\cos 2x=\frac{1+2\tan x-{\tan }^{2}x}{1+{\tan }^{2}x}$

• 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. Assertion is incorrect but Reason is correct

Answer 1

The correct option is D Assertion is incorrect but Reason is correct

U\sin g $\sin 2x=\frac{2\tan x}{1+{\tan }^{2}x}$ and $\cos 2x=\frac{1-{\tan }^{2}x}{1+{\tan }^{2}x}$
We get $\sin 2x+\cos 2x=\frac{2\tan x+1-{\tan }^{2}x}{1+{\tan }^{2}x}$
Hence reason is true
Now,
$2\sin 2x-\cos 2x=1\Rightarrow \frac{4\tan x-1+{\tan }^{2}x}{1+{\tan }^{2}x}=1\Rightarrow 4\tan x-1+{\tan }^{2}x=1+{\tan }^{2}x\Rightarrow \tan x=\frac{1}{2}$
Therefore
$\sin 2x+\cos 2x=\frac{2\tan x+1-{\tan }^{2}x}{1+{\tan }^{2}x}=\frac{1+1-\frac{1}{4}}{1+\frac{1}{4}}=\frac{7}{5}$
Hence assertion is false