Let f: R → R be defined as f(x + y) + f(x - y) = 2f(x)f(y),...

Let $f\left(x\right) = cos\lambdax$

$∵ f\left(\dfrac{1}{2}\right) = - 1$

So, $- 1 = cos\dfrac{\lambda}{2} ⇒ \dfrac{\lambda}{2} = \pi$

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

Now $k$ is a natural number

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

$\mathop{∑}_{k = 1}^{20}\dfrac{1}{\sin \left(k\right)\sin \left(k + 1\right)} = \dfrac{1}{\sin 1}\mathop{∑}_{k = 1}^{20}\left[\dfrac{\sin \left(\left(k + 1\right) - k\right)}{sink \cdot \sin \left(k + 1\right)}\right]$

$\dfrac{1}{\sin 1}\mathop{∑}_{k = 1}^{20}\left[\dfrac{\sin \left(k + 1\right)cosk - sinkcos\left(k + 1\right)}{sink \cdot \sin \left(k + 1\right)}\right]$

$= \dfrac{1}{\sin 1}\mathop{∑}_{k = 1}^{20}\left(cotk - \cot \left(k + 1\right)\right)$

$= \dfrac{\cot 1 - \cot 21}{\sin 1} = \dfrac{\dfrac{\cos 1}{\sin 1} - \dfrac{\cos 21}{\sin 21}}{\sin 1}$

$= \dfrac{\sin 21\cos 1 - \sin 1\cos 21}{\sin^{2}1\sin 21} = \sin \left(21 - 1\right){cosec}^{2}1cosec21$

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