Question

Modulus of $\frac{\cos \theta -i\sin \theta }{\sin \theta -i\cos \theta }$ is

• A. $0$
• B. $2\theta$
• C. $\pi -2\theta$
• D. None of these

Answer 1

Answer: (D)

None of these

Multiply By complex conjugated

$\frac{\cos \theta -i\sin \theta }{{\sin }^2\theta +{\cos }^2\theta }(\sin \theta +i\cos \theta )$

$\Rightarrow (\cos \theta -i\sin \theta )(\sin \theta +i\cos \theta )$

$\Rightarrow \cos \theta .\sin \theta +\sin \theta .\cos \theta +i{\cos }^2\theta -i{\sin }^2\theta -i{\sin }^2\theta$

$\Rightarrow 2\sin \theta .\cos \theta +i({\cos }^2\theta -{\sin }^2\theta )$

$\because 2\sin \theta .\cos \theta =\sin 2\theta$

and $co{s}^2\theta -{\sin }^2\theta =\cos 2\theta$

$\Rightarrow \sin 2\theta +i\cos 2\theta$

Now modules is $=\sqrt{{\sin }^{2}2\theta +{\cos }^{2}2\theta }$

$=\sqrt{1}$

$=1$

So, Ans is none of these.