Question

If\[cosp \theta +cosq \theta=0\], prove that the different values of \(\theta\) from two arithmetical progressions in which the common differences are\(\frac{2 \pi}{p+q}\) &\(\frac{2 \pi}{p-q}\).

Answer 1

@ganeshie8

Answer 2

hint:
we can rewrite such equation, by means of the Prosthaphaeresis formulas, as below:
\[\Large 2\cos \left( {\frac{{p + q}}{2}\theta } \right)\cos \left( {\frac{{p - q}}{2}\theta } \right) = 0\]
which is equivalent to these equations:
\[\Large \begin{gathered}
\cos \left( {\frac{{p + q}}{2}\theta } \right) = 0 \Rightarrow \theta = \frac{\pi }{{p + q}} + \frac{{2k}}{{p + q}}\pi ,\quad k \in \mathbb{Z} \hfill \\
\hfill \\
\cos \left( {\frac{{p - q}}{2}\theta } \right) = 0 \Rightarrow \theta = \frac{\pi }{{p - q}} + \frac{{2h}}{{p - q}}\pi ,\quad h \in \mathbb{Z} \hfill \\
\end{gathered} \]

Answer 3

I agree @Michele_laino sir....This is what i were also doing right now...

Answer 4

thanks! @samigupta8

Answer 5

THanks a lot @Michele_Laino sir