If sin θ + cos θ = 1, Find the Value of sin 2θ

Solution

Given,

\[ \sin\theta+\cos\theta=1 \]

Squaring both sides,

\[ (\sin\theta+\cos\theta)^2=1^2 \]

\[ \sin^2\theta+\cos^2\theta+2\sin\theta\cos\theta=1 \]

Using the identity:

\[ \sin^2\theta+\cos^2\theta=1 \]

we get

\[ 1+2\sin\theta\cos\theta=1 \]

\[ 2\sin\theta\cos\theta=0 \]

Using

\[ \sin2\theta=2\sin\theta\cos\theta \]

Therefore,

\[ \boxed{\sin2\theta=0} \]