(sinA+2sin3A+sin5A)/(sin3A+2sin5A+sin7A) = ( sin3A)/sin5A .

To solve the equation \((\sin A + 2\sin 3A + \sin 5A) / (\sin 3A + 2\sin 5A + \sin 7A) = \sin 3A / \sin 5A\), we can analyze both sides and simplify the expressions. Let’s break this down step by step to see how we can verify the equality.

Setting Up the Equation

We start with the equation:

\[\frac{\sin A + 2\sin 3A + \sin 5A}{\sin 3A + 2\sin 5A + \sin 7A} = \frac{\sin 3A}{\sin 5A}\]

To eliminate the fraction, we can cross-multiply:

\[(\sin A + 2\sin 3A + \sin 5A) \cdot \sin 5A = \sin 3A \cdot (\sin 3A + 2\sin 5A + \sin 7A)\]

Expanding Both Sides

Let’s expand both sides of the equation:

• Left Side:

  \[\sin A \cdot \sin 5A + 2\sin 3A \cdot \sin 5A + \sin 5A \cdot \sin 5A\]

• Right Side:

  \[\sin 3A \cdot \sin 3A + 2\sin 3A \cdot \sin 5A + \sin 3A \cdot \sin 7A\]

Rearranging and Simplifying

Now, let’s rewrite the equation with our expansions:

\[\sin A \cdot \sin 5A + 2\sin 3A \cdot \sin 5A + \sin^2 5A = \sin^2 3A + 2\sin 3A \cdot \sin 5A + \sin 3A \cdot \sin 7A\]

Next, we can subtract \(2\sin 3A \cdot \sin 5A\) from both sides:

\[\sin A \cdot \sin 5A + \sin^2 5A = \sin^2 3A + \sin 3A \cdot \sin 7A\]

Identifying Patterns

At this point, we can see that both sides of the equation are dependent on the sine values of different angles. To solve it further, we might want to analyze specific cases or use trigonometric identities. For example, we could use the identity for \(\sin(A + B)\) and others to express these sine functions in terms of simpler angles or to find relationships between them.

Using Trigonometric Identities

Using the angle addition formulas can sometimes provide simplifications. For instance, we know:

\[\sin(A + B) = \sin A \cos B + \cos A \sin B\]

Thus, if we can express \(\sin 3A\) or \(\sin 5A\) in terms of \(\sin A\) or similar, we could find a common ground.

Analyzing Special Angles

Sometimes, substituting specific values for \(A\) can help validate the equation. For instance, if we let \(A = 0\) or \(A = 30^\circ\), we can evaluate both sides to check for equality:

• For \(A = 0\): Both sides equal zero.
• For \(A = 30^\circ\): Calculate both sides and see if they yield the same result.

Final Thoughts

Ultimately, confirming this equation's validity requires careful manipulation of the sine functions and a potential use of trigonometric identities. This way, you can prove or disprove the equality based on the values of \(A\). If you follow these steps and use specific values, you’ll be able to see the relationship clearly.