Question

Which of the following relations is/are an identity?

• A. $\sin x^2=2\sin \frac{x^2}{2}\cos \frac{x^2}{2}$
• B. $\sin x=\frac{2\tan \frac{x}{2}}{1+{\tan }^2\frac{x}{2}}$
• C. $\log xy=\log |x|+\log |y|$
• D. $\sqrt{1-{\sin }^{2}x}=\cos x$

Answer 1

Answer: (A), (B)

$\sin x^2=2\sin \frac{x^2}{2}\cos \frac{x^2}{2}$
$\sin x=\frac{2\tan \frac{x}{2}}{1+{\tan }^2\frac{x}{2}}$

Option $A$ is an identity as $\sin 2x=2\sin x\cos x\Rightarrow \sin x^2=2\sin \frac{x^2}{2}\cos \frac{x^2}{2}$
Option $B$ is also Identity
Option $C$ is wrong as $\log xy$ is define only when $x>0$ and $y>0$ or $x<0$ and $y<0$. However, if $xy<0$, RHS will still be defined, hence not an identity.
Option $D$ is wrong as $\sqrt{1-{\sin }^{2}x}=\pm \cos x$
Hence, options $A$ and $B$ are correct.