Question

Prove.

Answer 1

\[\int_0^\pi \dfrac{\sin\left(n + \frac{1}{2}\right)x}{\sin \frac{x}{2}} = \pi\]

Answer 2

Induction lagega bhai?

Answer 3

Hmm! ek second.. check karta hun!

Answer 4

Induction se karke dekha tumne?

Answer 5

Hmm! dekhte hain... koi aur bhi tareeka hona toh chahiye

Answer 6

My book asks me to prove it through the relation\[\frac{\sin \left(n + \frac{1}{2}\right)x}{2\sin \frac{\pi}{2}} = 1 + \cos x + \cos 2x + \cdots + \cos n x\]

Answer 7

Phew! Never read this formula earlier. Will have to check it then.

Answer 8

Looks like AP wala formula.

Answer 9

Yeah, but if we go with the method provided, then its not that difficult. I'm thinking how did the book arrive to that relation.

Answer 10

In the relation you provided, is it 2 sin (x/2) or 2 sin (pi/2) ?

Answer 11

Sum when angles are in AP right?

Answer 12

Yeah, I think that is a misprint. It should be \(x/2\)

Answer 13

Well then, it should be easy now.

Answer 14

Yeah!

Answer 15

I also had an issue with the relation. Thanks.

Answer 16

Ab toh bas integrate hi karna hoga har term ko ..

Answer 17

(Y)

Answer 18

But ye relation ko prove karke dekho.. maza aayega!

Answer 19

@mathslover Sum of angles in AP hi hai na?