Statement-1: If 2sin2⁡(π/2)cos2⁡x=1−cos⁡(πsin⁡2x),x≠(2n+1)π/2,n is a integer,then sin⁡2x+cos⁡2x is equal to 1/5.Statement-2: sin⁡2x+cos⁡2x=2−(tan⁡x−1)21+tan2⁡x

sin⁡2x+cos⁡2x=2tan⁡x+1−tan2⁡x1+tan2⁡x=2−(tan⁡x−1)21+tan2⁡x⇒statement-2 is True.In statement -1,2 sin2⁡πcos2⁡x2=2sin2⁡πsin⁡2x2⇒ cos2⁡x=sin⁡2x⇒ cos⁡x(cos⁡x−2sin⁡x)=0⇒tan⁡x=1/2 as cos⁡x≠0.From statement-2, sin⁡2x+cos⁡2x=7/5⇒ statement-I is False.

Statement-1: If 2sin2⁡(π/2)cos2⁡x=1−cos⁡(πsin⁡2x),x≠(2n+1)π/2,n is a integer,then sin⁡2x+cos⁡2x is equal to 1/5.Statement-2: sin⁡2x+cos⁡2x=2−(tan⁡x−1)21+tan2⁡x

sin⁡2x+cos⁡2x=2tan⁡x+1−tan2⁡x1+tan2⁡x=2−(tan⁡x−1)21+tan2⁡x⇒statement-2 is True.In statement -1,2 sin2⁡πcos2⁡x2=2sin2⁡πsin⁡2x2⇒ cos2⁡x=sin⁡2x⇒ cos⁡x(cos⁡x−2sin⁡x)=0⇒tan⁡x=1/2 as cos⁡x≠0.From statement-2, sin⁡2x+cos⁡2x=7/5⇒ statement-I is False.

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Statement-1: If $2 \sin^{2} ((π / 2) \cos^{2} x) = 1 - \cos (π \sin 2 x), x \neq (2 n + 1) π / 2, n$ is a integer,
then $\sin 2 x + \cos 2 x$ is equal to 1/5.
Statement-2: $\sin 2 x + \cos 2 x = \frac{2 - (\tan x - 1)^{2}}{1 + \tan^{2} x}$

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STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1

STATEMENT-1 is False, STATEMENT-2 is True

STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1

STATEMENT-1 is True, STATEMENT-2 is False

answer is D.

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Detailed Solution

$\begin{array}{l} \sin 2 x + \cos 2 x = \\ \frac{2 \tan x + 1 - \tan^{2} x}{1 + \tan^{2} x} = \frac{2 - (\tan x - 1)^{2}}{1 + \tan^{2} x} \end{array}$