Answer
See explanations.
Work Step by Step
a. Differentiate the given identity: $sin2\theta=2sin\theta\ cos\theta$
We have $LHS=2cos2\theta$ and $RHS=2cos^2\theta-2sin^2\theta=2cos2\theta$. Thus, LHS=RHS and $cos2\theta=cos^2\theta-sin^2\theta$ is also an identity.
b. Differentiate the given identity: $cos2\theta=cos^2\theta-sin^2\theta$
We have LHS=$-2sin2\theta$ and RHS=$-2sin\theta\ sin\theta-2sin\theta\ cos\theta=-2sin2\theta$. Thus LHS=RHS and $sin2\theta=2sin\theta\ cos\theta$ is also an identity.
