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

Answer 1

$$\begin{gathered} \mathrm{Given},\sin \theta +\cos \theta =1 \\ \mathrm{By}\mathrm{squaring}\mathrm{both}\mathrm{sides}, \\ \mathrm{we}\mathrm{get},{\left(\sin \theta +\cos \theta \right)}^2={\left(1\right)}^2=1 \\ \mathrm{i}.\mathrm{e}.{\sin }^2\theta +{\cos }^2\theta +2\sin \theta \cos \theta =1 \\ \mathrm{i}.\mathrm{e}.1+2\sin \theta \cos \theta =1 \\ \left[\mathrm{Since}{\sin }^2\theta +{\cos }^2\theta =1\left(\mathrm{using}\mathrm{identity}\right)\right] \\ \mathrm{i}.\mathrm{e}.2\sin \theta \cos \theta =0 \\ \Rightarrow \sin 2\theta =0 \\ \mathrm{Hence},\mathrm{the}\mathrm{correct}\mathrm{answer}\mathrm{is}\mathrm{option}\mathrm{C}. \end{gathered}$$