• MHB
• Monoxdifly
Thank you!In summary, the conversation discusses proving the identity \frac{cos2x+cos2y}{sin2x-sin2y}=\frac1{tan(x-y)} by manipulating the LHS using trigonometric identities. The final result is cot(x-y) or \frac1{tan(x-y)}.

#### Monoxdifly

MHB
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.

Monoxdifly said:
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,

\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*}

castor28 said:
Hi Monoxdifly,

\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.

Last edited:

## 1. What is the proof for the equation (cos2x+cos2y)/(sin2x−sin2y)=1/tan(x−y)?

The proof for this equation is based on the trigonometric identities and the properties of the tangent function. It involves manipulating the given equation using these identities and simplifying it until both sides are equal.

## 2. How do you solve this equation for a specific value of x and y?

To solve this equation for a specific value of x and y, you would substitute the values into the equation and simplify it. The resulting value should be equal on both sides, proving the validity of the equation for those specific values.

## 3. Is this equation true for all values of x and y?

Yes, this equation is true for all values of x and y. This can be proven by substituting different values for x and y and showing that both sides of the equation remain equal.

## 4. What are the practical applications of this equation?

This equation is commonly used in mathematics and physics to solve problems involving trigonometric functions. It can also be used to simplify more complex equations involving trigonometric functions.

## 5. Can this equation be used to prove other trigonometric identities?

Yes, this equation can be used to prove other trigonometric identities by manipulating it using other identities and properties of trigonometric functions. It is a useful tool in proving and simplifying various trigonometric equations.