Mathematics - Functions Question with Solution | TestHub

Let $f\left(x\right)=\cos \lambda x$

$\because f\left(\frac{1}{2}\right)=-1$

So, $-1=\cos \frac{\lambda }{2}\Rightarrow \frac{\lambda }{2}=\mathrm{\pi }$

Thus $f\left(x\right)=\cos 2\pi x$

Now $k$ is a natural number

Thus $f\left(k\right)=1$

$\sum _{k=1}^{20}\frac{1}{\sin \left(k\right)\sin \left(k+1\right)}=\frac{1}{\sin 1}\sum _{k=1}^{20}\left[\frac{\sin \left(\left(k+1\right)-k\right)}{\sin k·\sin \left(k+1\right)}\right]$

$\frac{1}{\sin 1}\sum _{k=1}^{20}\left[\frac{\sin \left(k+1\right)\cos k-\sin k\cos \left(k+1\right)}{\sin k·\sin \left(k+1\right)}\right]$

$=\frac{1}{\sin 1}\sum _{k=1}^{20}\left(\cot k-\cot \left(k+1\right)\right)$

$=\frac{\cot 1-\cot 21}{\sin 1}=\frac{{\displaystyle \frac{\cos 1}{\sin 1}}-{\displaystyle \frac{\cos 21}{\sin 21}}}{\sin 1}$

$=\frac{\sin 21\cos 1-\sin 1\cos 21}{{\sin }^{2}1\sin 21}=\sin \left(21-1\right){\mathrm{cosec}}^{2}1\mathrm{cosec}21$

$={cosec}^2\left(1\right)cosec\left(21\right)\sin \left(20\right)$