If \( \sin\theta+\cos\theta=1 \), Find the Value of \( \sin2\theta \)
Question
If
\[ \sin\theta+\cos\theta=1, \]
then the value of
\[ \sin2\theta \]
is
(a) \(1\)
(b) \(\frac12\)
(c) \(0\)
(d) \(-1\)
Solution
Given,
\[ \sin\theta+\cos\theta=1 \]
Squaring both sides,
\[ (\sin\theta+\cos\theta)^2=1 \]
\[ \sin^2\theta+\cos^2\theta+2\sin\theta\cos\theta=1 \]
Using
\[ \sin^2\theta+\cos^2\theta=1 \]
\[ 1+2\sin\theta\cos\theta=1 \]
\[ 2\sin\theta\cos\theta=0 \]
But
\[ \sin2\theta=2\sin\theta\cos\theta \]
Therefore,
\[ \sin2\theta=0 \]
Final Answer
\[ \boxed{0} \]
Hence, the correct option is (c) 0.