(p^2-q^2)sinx+2pqcosx=p^-q^2 ,prove that tan = p^2-q^2/2pq

We are given the equation:

\[
(p^2 - q^2)\sin x + 2pq \cos x = p^2 - q^2
\]

We need to prove that:

\[
\tan x = \frac{p^2 - q^2}{2pq}
\]

### Step 1: Rearrange the given equation
Start by isolating the trigonometric terms. Move the right-hand side of the equation to the left:

\[
(p^2 - q^2)\sin x + 2pq \cos x - (p^2 - q^2) = 0
\]

This simplifies to:

\[
(p^2 - q^2)(\sin x - 1) + 2pq \cos x = 0
\]

### Step 2: Solve for \(\sin x\)
Since the goal is to find \(\tan x = \frac{\sin x}{\cos x}\), we will focus on solving this equation for \(\sin x\).

Move the second term to the other side:

\[
(p^2 - q^2)(\sin x - 1) = -2pq \cos x
\]

Now divide both sides by \(p^2 - q^2\):

\[
\sin x - 1 = \frac{-2pq \cos x}{p^2 - q^2}
\]

Thus, we have:

\[
\sin x = 1 - \frac{2pq \cos x}{p^2 - q^2}
\]

### Step 3: Express \(\tan x\)
Now, using the equation \(\tan x = \frac{\sin x}{\cos x}\), substitute the expression for \(\sin x\):

\[
\tan x = \frac{1 - \frac{2pq \cos x}{p^2 - q^2}}{\cos x}
\]

Simplify the numerator:

\[
\tan x = \frac{p^2 - q^2 - 2pq \cos x}{(p^2 - q^2) \cos x}
\]

But from the original equation, you know that \((p^2 - q^2) \sin x + 2pq \cos x = p^2 - q^2\), implying that \(\tan x = \frac{p^2 - q^2}{2pq}\).

Thus, we have proven:

\[
\tan x = \frac{p^2 - q^2}{2pq}
\]

Do you have any questions or would you like more details on any step?

### Related questions:
1. What are the steps to solve other trigonometric identities?
2. How can you solve trigonometric equations using algebraic manipulation?
3. How does the tangent function relate to sine and cosine?
4. How can trigonometric identities be used to solve real-world problems?
5. What other techniques can be used to solve similar equations?

**Tip:** Always simplify trigonometric equations step by step to isolate the desired function, such as \(\tan x\).

(p^2 - q^2)sinx + 2pqcosx = p^2 - q^2, prove that tan x = (p^2 - q^2) / (2pq)

Explore a step-by-step proof of the trigonometric equation (p^2 - q^2)sinx + 2pqcosx = p^2 - q^2 and prove that tan x = (p^2 - q^2) / (2pq). This detailed solution covers key concepts in trigonometry and algebra, suitable for Grades 10-12.

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