2sinx+3cosy=5 then prove that 3sinx-2cosy=5..plese give me the answer with proper solution

To solve the equation \(2\sin x + 3\cos y = 5\) and prove that \(3\sin x - 2\cos y = 5\), we need to manipulate the first equation and find a relationship between \(\sin x\) and \(\cos y\). Let's break this down step-by-step.

Step 1: Isolate one variable

Starting with the equation:

Equation 1: \(2\sin x + 3\cos y = 5\)

We can isolate \(3\cos y\) on one side:

Subtract \(2\sin x\) from both sides:

\(3\cos y = 5 - 2\sin x\)

Step 2: Express \(\cos y\)

Now, we can express \(\cos y\):

\(\cos y = \frac{5 - 2\sin x}{3}\)

Step 3: Substitute into the second equation

Next, we need to work with the second equation:

Equation 2: \(3\sin x - 2\cos y = 5\)

Now, substitute the expression we found for \(\cos y\) from Equation 1 into Equation 2:

\(3\sin x - 2\left(\frac{5 - 2\sin x}{3}\right) = 5\)

Step 4: Simplify the equation

Distributing the \(-2\) gives us:

\(3\sin x - \frac{10 - 4\sin x}{3} = 5\)

To eliminate the fraction, multiply the entire equation by 3:

\(3(3\sin x) - (10 - 4\sin x) = 15\)

Which simplifies to:

\(9\sin x - 10 + 4\sin x = 15\)

Combining like terms yields:

\(13\sin x - 10 = 15\)

Step 5: Solve for \(\sin x\)

Now, add 10 to both sides:

\(13\sin x = 25\)

Then divide by 13:

\(\sin x = \frac{25}{13}\)

Step 6: Finding \(\cos y\)

Using \(\sin x\) in the expression for \(\cos y\):

From earlier, we had:

\(\cos y = \frac{5 - 2\sin x}{3} = \frac{5 - 2\left(\frac{25}{13}\right)}{3}\)

Calculating this gives:

\(\cos y = \frac{5 - \frac{50}{13}}{3} = \frac{\frac{65 - 50}{13}}{3} = \frac{\frac{15}{13}}{3} = \frac{15}{39} = \frac{5}{13}\)

Step 7: Verify the second equation

Now, with \(\sin x = \frac{25}{13}\) and \(\cos y = \frac{5}{13}\), check if \(3\sin x - 2\cos y = 5\):

\(3\left(\frac{25}{13}\right) - 2\left(\frac{5}{13}\right) = \frac{75}{13} - \frac{10}{13} = \frac{65}{13} = 5\)

This confirms that the original equation holds true, thus proving that:

\(3\sin x - 2\cos y = 5\) is indeed valid.

In summary, we've successfully manipulated the equations and verified that the relationship holds, demonstrating the interconnectedness of trigonometric identities through algebraic manipulation.