Question

If $\sin \theta +\cos \theta =\sqrt{2}$, find the value of $\sin \theta \times \cos \theta$.

• A. $2$
• B. $\sqrt{2}-1$
• C. $\frac{1}{2}$
• D. $2\sec \theta$

Answer 1

Answer: (C)

$\frac{1}{2}$

$\sin \theta +\cos \theta =\sqrt{2}$

Squaring both sides, we get

$\Rightarrow {(\sin \theta +\cos \theta )}^2=2$

$\Rightarrow {\sin }^2\theta +{\cos }^2\theta +2\sin \theta .\cos \theta =2$

$\Rightarrow 1+2\sin \theta .\cos \theta =2$

$\Rightarrow 2\sin \theta \cos \theta =1$

$\Rightarrow \sin \theta \cos \theta =\frac{1}{2}.$

Hence, the answer is $\frac{1}{2}.$