Question

$\tiny\sf \pink{\sin( \frac{\pi}{2020} ) + \sin( \frac{3\pi}{2020} ) + \sin( \frac{5\pi}{2020} ) + ... + \sin( \frac{2019\pi}{2020} ) = \csc( \frac{\pi}{2020} ) }$​

Answer 1

Answer:

Step-by-step explanation:

We have,

$\mapsto\,\,\tt{\green{S=sin\left(\dfrac{\pi}{2020}\right)+sin\left(\dfrac{3\pi}{2020}\right)+sin\left(\dfrac{5\pi}{2020}\right)+\cdots+sin\left(\dfrac{2019\pi}{2020}\right)}}$

$\tt{\implies\,S=sin\left(\dfrac{\pi}{2020}\right)+sin\left(\dfrac{\pi}{2020}+\dfrac{2\pi}{2020}\right)+sin\left(\dfrac{\pi}{2020}+\dfrac{4\pi}{2020}\right)+\cdots+}\\\tt{sin\left(\dfrac{\pi}{2020}+\dfrac{2018\pi}{2020}\right)}$

$\bigstar\,\,\bf{Put\,\,\,\dfrac{\pi}{2020}=\theta\,\,\,\,\,and\,\,\,\,\,\dfrac{2\pi}{2020}=\phi}$

So,

$\tt{S=sin\left(\theta\right)+sin\left(\theta+\phi\right)+sin\left(\theta+2\phi\right)+\cdots+sin\left(\theta+1009\phi\right)}$

$\tt{\implies\,S=sin\left(\theta\right)+sin\left(\theta+\phi\right)+sin\left(\theta+2\phi\right)+\cdots+sin\left\{\theta+(1010-1)\phi\right\}}$

$\sf{We\,\,\,know\,,}\\\\\boxed{\bf{\green{sin(a)+sin(a+h)+sin(a+2h)+\cdots+sin\{a+(n-1)h\}}}}}\\\\\boxed{\bf{\green{=\dfrac{sin\left\{a+\left(\dfrac{n-1}{2}\right)h\right\}\cdot\,sin\left(\dfrac{nh}{2}\right)}{sin\left(\dfrac{h}{2}\right)}}}}}$

So,

$\tt{\implies\,S=\dfrac{sin\left\{\theta+\left(\dfrac{1010-1}{2}\right)\phi\right\}\cdot\,sin\left(\dfrac{1010\phi}{2}\right)}{sin\left(\dfrac{\phi}{2}\right)}}$

Now, putting the value of $\theta$ and $\phi$

$\tt{\implies\,S=\dfrac{sin\left\{\dfrac{\pi}{2020}+\dfrac{1009}{2}\cdot\dfrac{2\pi}{2020}\right\}\cdot\,sin\left(\dfrac{1010}{2}\cdot\dfrac{2\pi}{2020}\right)}{sin\left(\dfrac{1}{2}\cdot\dfrac{2\pi}{2020}\right)}}$

$\tt{\implies\,S=\dfrac{sin\left(\dfrac{\pi}{2020}+\dfrac{1009\pi}{2020}\right)\cdot\,sin\left(\dfrac{\pi}{2}\right)}{sin\left(\dfrac{\pi}{2020}\right)}}$

$\tt{\implies\,S=\dfrac{sin\left(\dfrac{1010\pi}{2020}\right)\cdot\,1}{sin\left(\dfrac{\pi}{2020}\right)}}$

$\tt{\implies\,S=\dfrac{sin\left(\dfrac{\pi}{2}\right)\cdot\,1}{sin\left(\dfrac{\pi}{2020}\right)}}$

$\tt{\implies\,S=\dfrac{1\cdot\,1}{sin\left(\dfrac{\pi}{2020}\right)}}$

$\tt{\implies\,S=\dfrac{1}{sin\left(\dfrac{\pi}{2020}\right)}}$

$\tt{\implies\,S=cosec\left(\dfrac{\pi}{2020}\right)}$