Question:
If sin θ + cos θ = 1, then the value of sin 2θ is equal to
(a) 1
(b) $\frac{1}{2}$
(c) 0
(d) –1
Solution:
Given, $\sin \theta+\cos \theta=1$
By squaring both sides,
we get, $(\sin \theta+\cos \theta)^{2}=(1)^{2}=1$
i.e. $\sin ^{2} \theta+\cos ^{2} \theta+2 \sin \theta \cos \theta=1$
i. e. $1+2 \sin \theta \cos \theta=1$
Since $\sin ^{2} \theta+\cos ^{2} \theta=1$ (using identity)
1.e. $2 \sin \theta \cos \theta=0$
$\Rightarrow \sin 2 \theta=0$
Hence, the correct answer is option C.
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