Question

Find $cos(\frac{\pi }{2}+\theta )+cos(\frac{5\pi }{2}+\theta )+cos(\frac{7\pi }{2}+\theta )\cdots$ upto $2021$ terms.

Answer 1

$\cos (\frac{\pi }{2}+\theta )=-\sin \theta \cos (\frac{5\pi }{2}+\theta )=+\sin \theta \cos (\frac{5\pi }{2}+\theta )=\sin \theta ....$

$\to$ Terms are identical apart from alternating sign.

$\to$ There is odd number of terms; $2$ each adjacent pair sums to zero.

$\to \therefore$ Sum of $2021$ terms$=$ the first term$=-\sin \theta$