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Trigonometry [ASK} Prove (cos2x+cos2y)/(sin2x−sin2y)=1/tan(x−y)

Monoxdifly

Well-known member
Aug 6, 2015
267
Prove that \(\displaystyle \frac{cos2x+cos2y}{sin2x-sin2y}=\frac1{tan(x-y)}\). Can someone provide me some hints? I tried to manipulate the right-hand expression but got back to square one.
 

castor28

Well-known member
MHB Math Scholar
Oct 18, 2017
235
Brussels, Belgium
Prove that \(\displaystyle \frac{cos2x+cos2y}{sin2x-sin2y}=\frac1{tan(x-y)}\). Can someone provide me some hints? I tried to manipulate the right-hand expression but got back to square one.
Hi Monoxdifly ,

You could start with the LHS and the identities:

\begin{align*}
\cos a + \cos b &= 2\cos\frac{a+b}{2}\cos\frac{a-b}{2}\\
\sin a - \sin b &= 2\sin\frac{a-b}{2}\cos\frac{a+b}{2}
\end{align*}
 

Monoxdifly

Well-known member
Aug 6, 2015
267
Hi Monoxdifly ,

You could start with the LHS and the identities:

\begin{align*}
\cos a + \cos b &= 2\cos\frac{a+b}{2}\cos\frac{a-b}{2}\\
\sin a - \sin b &= 2\sin\frac{a-b}{2}\cos\frac{a+b}{2}
\end{align*}
Ah, let's see...
\(\displaystyle \frac{cos2x+cos2y}{sin2x-sin2y}\)=\(\displaystyle \frac{2cos\frac{2x+2y}{2}cos\frac{2x-2y}{2}}{2sin\frac{2x-2y}{2}cos\frac{2x+2y}{2}}\)=\(\displaystyle \frac{cos(x-y)}{sin(x-y)}\)= cot(x - y) = \(\displaystyle \frac1{tan(x-y)}\)
Wew. Just 4 steps.
 
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