<p>Let <img alt="$x$" height="10" src="https://latex.artofproblemsolving.com/2/6/e/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png" width="12" /> and <img alt="$y$" height="13" src="https://latex.artofproblemsolving.com/0/9/2/092e364e1d9d19ad5fffb0b46ef4cc7f2da02c1c.png" width="11" /> be real numbers such that <img alt="$\frac{\sin x}{\sin y} = 3$" height="40" src="https://latex.artofproblemsolving.com/d/e/2/de209176871ea05ad0759c6aa78048bf48cfb1bf.png" width="72" /> and <img alt="$\frac{\cos x}{\cos y} = \frac12$" height="40" src="https://latex.artofproblemsolving.com/d/7/5/d7598cd5337224ab03d83e92be673996720b1e11.png" width="77" />. The value of <img alt="$\frac{\sin 2x}{\sin 2y} + \frac{\cos 2x}{\cos 2y}$" height="40" src="https://latex.artofproblemsolving.com/7/8/0/780c230da33340a3e9edf770053ab8638320a3a7.png" width="120" /> can be expressed in the form <img alt="$\frac pq$" height="36" src="https://latex.artofproblemsolving.com/8/c/0/8c08adadeb0c9372654ee5cc432ccb32f255443b.png" width="11" />, where <img alt="$p$" height="13" src="https://latex.artofproblemsolving.com/3/6/f/36f73fc1312ee0349b3f3a0f3bd9eb5504339011.png" width="11" /> and <img alt="$q$" height="13" src="https://latex.artofproblemsolving.com/0/6/1/0615acc3725de21025457e7d6f7694dab8e2f758.png" width="11" /> are relatively prime positive integers. Find <img alt="$p+q$" height="14" src="https://latex.artofproblemsolving.com/1/7/0/170200756543596efeb46fca1cf29a617e8c9192.png" width="41" />.</p>

Question No: 462

Let $x$ and $y$ be real numbers such that $\frac{\sin x}{\sin y} = 3$ and $\frac{\cos x}{\cos y} = \frac12$ . The value of $\frac{\sin 2x}{\sin 2y} + \frac{\cos 2x}{\cos 2y}$ can be expressed in the form $\frac pq$ , where $p$ and $q$ are relatively prime positive integers. Find $p+q$ .

Editorial

Tags: Trigonometry

Question Details