Question

Find the value of $\sin 2x+\cos 2x$ if $0\le x\le \frac{\pi }{2}$, ${\sin }^{2}x=a$ and ${\cos }^{2}x=b$.

• A. $2\sqrt{ab}+b-a$
• B. $2ab+a^2-b^2$
• C. $2\sqrt{ab}+2a-1$
• D. $2ab+2a-1$
• E. $2\sqrt{ab}-2b+1$

Answer 1

The correct option is A $2\sqrt{ab}+b-a$

Given, ${\sin }^{2}x=a,{\cos }^{2}x=b$

$\Rightarrow {\sin }^{2}x=a$, which implies $\sin x=\sqrt{a}$
$\Rightarrow {\cos }^{2}x=b$, which implies $\cos x=\sqrt{b}$
$\Rightarrow \sin 2x+\cos 2x=2\sin x\cos x+1-2{\sin }^{2}x$

$=2\sin x\cos x+1-{\sin }^{2}x+{\sin }^{2}x=2\sin x\cos x+{\cos }^{2}x+{\sin }^{2}x$

$=2\sqrt{ab}+b-a$